The Classification of the First Order Ordinary Differential Equations With

The Classification of the First Order Ordinary Differential Equations with the PainlevéProperty

The Classical and a Modern Algebro-Geometric Approach

Version 1.0 Georg Muntingh Preface and Acknowledgments

I would like to acknowledge my considerable debt to many people who helped me. First of all I want to thank my supervisors, Marius van der Put and Jaap Top. Their doors were always open, and they never seemed to tire of ex- plaining something to me for the second or even the third time, encouraging me to keep trying. Secondly my gratitude goes out to Professor Masahiko Saito from Kobe University for his talk in Utrecht that indirectly led to this thesis and his help with references for first order equations. Several people are responsible for making the text more readable and clear, both on the mathematical and on the linguistical part. For that my thanks go to Monique van Beek, Jeroen Sijsling, Laurens van der Starre and of course my girlfriend Annett. Moving to Oslo to live with her formed the primary motivation to finish up my thesis. Furthermore I am of course indebted to my family for supporting me all my life. I would like to thank several people who provided me with the facilities I needed to write this thesis, in particular the system operators Harm Paas, Jurjen Bokma and Peter Arendz for the GNU/Linux Debian system at the Department of Mathematics at the Rijksuniversiteit Groningen, the clean- ing lady Anja who was always cheerful in the morning and Ineke from the administration for the invigorating chats and the many cups of coffee. For typesetting I used LATEX 2ε and the very convenient LATEX editor Kile. The pictures were made with The Gimp, Gnuplot and Dia, and the frontispiece was inspired by the logo of Wikipedia. The first chapter will serve as an introduction to PainlevéTheory, giving some motivation and intuitive definitions. At the end of the chapter several questions will be posed that will be discussed later in the thesis. The second chapter will deal with a large part of the mathematics that is needed later on, especially in the chapter on modern theory. In the third chapter an

i overview of the historical development of PainlevéTheory will be given, together with rewritten classical theorems and a rewritten classical proof. After that, in the fourth chapter, a detailed modern theory will be presented, followed by the fifth and final chapter containing conclusions and suggestions for future work. At the end of the document one can find an index of some terminology and names, referring to the page where they occurred first, and a list of symbols accompanied by a short description.

ii CONTENTS

1 Introduction 2 1.1 Problematic Points ...... 2 1.2 The Painlev´eProperty ...... 4 1.3 First Order Equations with the PP ...... 6

2 Prerequisites 9 2.1 Fields of Functions ...... 9 2.2 Factorizing a Polynomial Differential Equation ...... 12 2.3 Local Rings and Valuations ...... 13 2.4 Ramification and Branch Points ...... 15 2.5 Differential Function Fields ...... 16

3 Classical Painlev´eTheory 19 3.1 First Order Painlev´eTheory in the Literature ...... 19 3.2 Classical Proof of the Theorem of Briot and Bouquet . . . . . 24 3.3 The Algorithm Indicated by the Theorem of Briot and Bouquet 31 3.4 Fuchs’s Criterion ...... 32

4 Modern PainlevéTheory 34 4.1 The Setting and Theme ...... 34 4.2 The Algebraic PainlevéProperty ...... 37 4.3 The Riccati Equation ...... 38 4.4 The Generalized Weierstrass Equation ...... 40 4.5 Classification of the First Order Autonomous Equations . . . 42 4.6 Classification of the First Order Equations ...... 46

5 Discussion 57 5.1 Conclusions ...... 57

iii 5.2 Future Work ...... 58

A Implementation of a Painlev´eTest in Maple 59

B The Painlev´eProperty in Physics 61

List of Symbols 67

iv LIST OF FIGURES

1 A portrait of Paul Painlev´efrom 1929 ...... 1

1.1 A picture of the Riemann surface corresponding to the logarithm, above a disk in the complex plane...... 3

2.1 Several structures of functions together with their relations. . 11

3.1 Two 19th century mathematicians who layed the foundations for PainlevéTheory...... 20 3.2 Two mathematicians who solved the general case for first order equations...... 22 3.3 Two mathematicians who redid some classical PainlevéTheory. 24

4.1 A schematic representation of the situation in the proof of Proposition 4.28...... 50

B.1 A caricature of Paul Painlev´efrom 1932 ...... 69

v Figure 1: A portrait of Paul Painlev´efrom 1929

1 CHAPTER 1 Introduction

Les Mathématiques constituent un continent solidement agencé, dont tous les pays sont bien reliésles uns aux autres; l’oeuvre de Paul Painlevéest une ˆıle originale et splendide dans l’océan voisin,– H. Poincaré

he class of all differential equations is enormous and very complicated T to study in general. The best one can do is to restrict our research to a class of differential equations that is easy enough to say sensible things about and wide enough to describe a wide spectrum of phenomena. This thesis is about such a class, namely the class of first order ordinary differential equations with the PainlevéProperty. Before we define the PainlevéProperty, we give in Section 1.1 some conceptual definitions of the problematic points of a differential equation. Armed with these notions, we give a conceptual definition of the Painlevé Property in Section 1.2 and motivate why we should study equations with this property. Finally, in Section 1.3, we shall restrict ourselves to the case of first order ordinary differential equations and discuss what kind of questions we can ask ourselves. The remainder of the thesis will then be concerned with the answers to these questions.

1.1 Problematic Points

In this section, we will look at some of the problems that can occur regarding the solutions of differential equations. To be more precise, we shall introduce three types of so-called problematic points. When we know which problems can occur in this context, we can restrict ourselves to differential equations that do not have these problems and try to examine this simplified case. This is exactly what is done in PainlevéTheory.

2 Figure 1.1: A picture of the Riemann surface corresponding to the logarithm, above a disk in the complex plane.

An example of a problematic point is a so called branch point. This is a point in which multivaluedness of the solution occurs. This means that there does not exist a neighborhood of the point in the complex plane in which we can define a solution, but there does exist a neighborhood in the universal covering space on which we can. Let us illustrate this with an example. Example 1.1 (Branch point of the logarithm). Consider the initial value problem 1 f 0(z) = , f(1) = 0. z In a neighborhood of the point 1, this equation clearly has a solution f(z) := log z. From function theory, however, we know that we cannot get a power series solution in a neighborhood of 0. Furthermore, we cannot extend the solution in a neighborhood of 1 to a function that has a power series expansion in a punctured neighborhood of 0. So what can we do? The answer is that we can construct a larger space, called a Riemann surface, on which a solution can be defined. Maximal solutions that we were able to define on the complex plane, for instance the analytic function log : C\R≤0 → C that restricts to the ordinary logarithm on R>0, then appear as projections to C of restrictions to a certain sheet of our grand solution on the Riemann surface (see figure 1.1). In general we call a point a branch point of the differential equation if it has a punctured neighborhood U in which a power series solution of the differential equation can everywhere determined locally, yet there exists no

3 global solution on U restricting to a power series everywhere, no matter how small U is chosen. Another problematic point is a pole. This is a point for which the solution goes to infinity when we approach this point, but there exists a monomial zk such that the product of zk and the solution can locally be expressed as a convergent power series. Another way to say this is that the function has a pole in 0 if it can locally be expressed as a Laurent series in z. An easy example of a differential equation with a pole is the following. Example 1.2 (Pole). Consider the initial value problem 1 f 0(z) = − , f(1) = 1. z2 1 This initial value problem clearly has a unique solution f(z) := z , which goes to infinity at z = 0. If we multiply this solution by z1, then we obtain the function that equals 1 in every point. This function is clearly the convergent power series 1 + 0 · z + 0 · z2 + ··· . Therefore this initial value problem has a pole at z = 0. Another type of singularities is the type of essential singularities. These are much worse than poles. Example 1.3 (Essential singularity). Consider the initial value problem 1 f 0(z) = − f(z), f(1) = e. z2 This equation has a unique solution f(z) := e1/z, which has a singularity at z = 0. Since there does not exist a natural number k such that zke1/z is a convergent power series at z = 0, this is an essential singularity. We shall call all these three types of points problematic points. The last two, where the function becomes infinity, are called singularities.

1.2 The Painlev´eProperty

In the previous subsection we discussed what kind of problematic points can arise in solutions of differential equations. In this subsection these points will be the ingredients of our definition of the PainlevéProperty. We say that a problematic point of an initial value problem is movable, if it changes position when we (slightly) change the initial condition (that is, if we change the f0 in the initial condition f(z0) = f0, and no matter how small this change is). If a problematic point is not movable, then we say it is fixed. Definition 1.4 (PainlevéProperty). A differential equation has the PainlevéProperty (PP) if it has no movable branch points and no movable essential singularities.

4 Remark 1.5 (Why poles are not that bad). Note that fixed problematic points and even movable poles are allowed for a differential equation with the PP. The reason for this is that the natural object for differential equations to live upon is not the complex plane, but the Riemann sphere P. This is the complex plane together with an additional point at infinity, and it can be drawn as a sphere whose top represents the point infinity and whose bottom represents the point zero. If we are interested in local properties, as we are with local solutions of the differential equation, then we only have to look at a piece of the Riemann sphere (a chart). Switching between the top chart and the bottom chart then comes down to substituting z−1 for z. So if a solutions has a pole at zero, then we can write it locally as a Laurent series, and if we switch to the chart at infinity we obtain a nice power series. Example 1.6. The problematic points of the Examples 1.1, 1.2 and 1.3 are all independent of the value of the initial condition. Therefore the differential equations from these examples have the PP. where f (n)(x) denotes the functions resulting after taking the derivative n times Remark 1.7. Linear differential equations are equations of the form

0 (n) a0(x)f(x) + a1(x)f (x) + ··· + an(x)f (x) = b(x), where f (n)(x) denotes the function resulting after taking the derivative of f(x) n times. The equations of the Examples 1.1, 1.2 and 1.3 are all linear, and they have the PP. The set of linear diﬀerential equations has been studied extensively, and one can show that they all have the PP. Remark 1.8. Though it is true that many equations in physics can be approx- imated by linear equations, most of them are highly nonlinear. Therefore the need arises to study a class of equations that captures a wide spectrum of these nonlinear phenomena but is still mathematically easy enough to say sensible things about. The equations satisfying the PP seem to be highly appropriate for this. In physics, there are a lot of models given by an equation with the PP. For a list of areas in physics in which the PP occurs, see Appendix B. This huge list suggests that the PP is a frequently occurring notion in physics, indicating that it has some physical meaning. In this sense, the equations with the PP form a very important class to study. Example 1.9. As an example, we ask ourselves for which positive integers p and q the equation

f 0p = f q, gcd(p, q) = 1 (1.1) has the Painlev´eProperty. We can restrict to initial conditions in the point 0, because the equation is autonomous. That is, the equation depends only on f 0 and f and not on

5 z. Suppose that we have a solution w of this equation for an initial condition w(0) = w0 6= 0. Thenin a neighborhood of 0 w satisfies one of the initial value problems 0 k q/p w = ζp w , w(0) = w0 6= 0, k = 0, . . . , p − 1, (1.2) p where ζp satisfies the equation ζp = 1. Therefore we can find all local solutions of (1.1) by finding all solutions of (1.2). Now what do these solutions look like? Let us try to find some solutions of (1.2). Separating variables and integrating gives us Z p p−q z(w) = ζ−kw−q/pdw + const = ζ−k w p + const. p p p − q Inverting this function, we find solutions p p − q p−q w(z) = ζk (z − const) . p p

Combining this with the initial condition w(0) = w0 6= 0, we find a solution p p − q p−q p−q w (z) = ζk z + w p k p p 0 to this initial value problem, which is analytic in a sufficiently small neighborhood of 0. For certain values of p and q it has a branch point at p−q −k p p z = −ζp p−q w0 , depending on the initial condition w(0) = w0. Using that gcd(p, p − q) = 1, we find that the criterion for the absence of (movable) branch points becomes: f 0p = f q has no movable branch points ⇐⇒ |p − q| = 1. What about the initial condition w(0) = 0? This is the remaining case for which the differential equation might have solutions. Such a solution is either the zero function (which has of course no branch points or essential singularities) or a function that is in some point unequal to 0. However, then it is a solution to an initial value problem with w(0) 6= 0 as well, in which case it has no movable branch points or essential singularities.

1.3 First Order Equations with the PP

Having defined the PP for any differential equation in the previous section, we shall from now on restrict ourselves to first order ordinary differential equations. More precisely, we shall in the remainder of the thesis consider differential equations of the form F (z, y0, y) = 0, with F a rational function of y0 and y and some algebraic function of z. We can then formulate the following questions.

6 (a) The mathematician Ja- (b) The German mathemati- copo Francesco Riccati from cian Karl Theodor Wilhelm the Venetian Republic (1676- Weierstrass (1815-1897). 1754).

1. How can we determine if such an equation has the PP? 2. Can we give a precise classification of all equations of this form with the PP? 3. If we can give such a classification, how can we, given a differential equation, effectively determine which of the categories of the classification it belongs to? 4. If we can give such a classification, can we, given a differential equation with the PP, effectively determine what kind of transformation leads to the canonical form in its category? 5. Can we give a modern equivalent of the PP?

It is this type of questions that this thesis tries to answer. The answer to the first question is known in the classical literature as Fuchs’s Criterion, and we shall discuss it in Sections 3.1 and 3.4. After we give a modern description of first order PainlevéTheory in Chapter 4, we can also obtain an answer from standard algorithms in algebraic geometry. Question (2) was completely solved in a classical way at the end of the 19th century [17], and in Chapter 4 we answer the question in a more modern way using some differential algebraic geometry. It turns out that by means of appropriate transformations we can transform such an equation with the PP into either a Riccati equation y0 = a(z)y2 + b(z)y + c(z),

7 or a generalized Weierstrass equation,

02 3 y = a(z)(4y − g2y − g3).

But this classification directly gives rise to question (3). Again after having obtained a modern description, it becomes clear that the differential equations are classified by the genus of a certain associated curve, and then standard algorithms from geometry give the answer. The answer to question (4) was written down very precisely by Malmquist [14] but probably also already by Painlevé[17]. The modern answer to this question will be that each differential equation reduces to one of the categories by means of a transformation that induces an isomorphism on the associated differential function field. This will be made somewhat more clear in a remark at the end of Section 4.1. The final question was already answered by Matsuda [15] but without any motivation. In the Sections 4.5 and 4.6 it will be derived that an equation has the PP if and only if the “derivation on the associated function field is regular.” What we precisely mean by this will become clear in the last section of the next chapter.

8 CHAPTER 2 Prerequisites

n this chapter we shall discuss the prerequisites for the remainder of the I thesis not taught in standard undergraduate courses. We assume that the reader is familiar with some notions encountered in a first course in commutative algebra, algebraic geometry and function theory. We start in the first section with looking at several structures of functions that we shall frequently use throughout the thesis. In the next section these notions will be applied to find an explicit description for the factorization of a polynomial differential equation. This will be used in the next chapter. After this, two short sections on local theory follow, at the end of which we are able to give a precise definition of ramification (and thereby branch points). The final section is about function fields with a derivation on it. Such differential function fields will replace the notion of a differential equation in Chapter Modern PainlevéTheory.

2.1 Fields of Functions

Throughout the thesis we shall consider several types of functions, such as polynomials, power series etc. Each of these types of functions forms a class that carries some algebraic structure, and these algebraic structures themselves are related to each other by constructions from algebra. In this section we shall describe the functions we need, describe the algebraic structures they belong to and describe the connection between these algebraic structures. Let R be a ring and k be a ﬁeld. We start by giving examples of rings and ﬁelds whose elements can be interpreted as (or more precisely induce) functions. In the rest of the thesis R and k will often denote the complex numbers, so we could take R = k = C in the example below. Moreover

9 every ring encountered in this thesis has characteristic zero. Example 2.1. (1) A polynomial in the variable X with coeﬃcients in R is an expression of the form

n a0 + a1X + ··· + anX , n ∈ N0, ai ∈ R, and all such polynomials together form the ring R[X] of polynomials with coefficients in R. (2) A rational function in the variable X with coefficients in R is an expression of the form n a0 + a1X + ··· + anX m , n ∈ N0, ai, bi ∈ R, b0 + b1X + ··· + bmX and all such rational functions together form the field R(X) of rational functions with coefficients in R. (3) An algebraic function in the variable X over k[X] is a function f(X) defined on some domain of k that satisfies a polynomial equation

n 0 = P0(X) + P1(X)f(X) + ··· + Pn(X)f(X) ,Pi(X) ∈ k[X], and all such algebraic functions together form the ﬁeld k(X) of algebraic functions over k[X]. (4) A formal power series in the variable X with coeﬃcients in R is an expression of the form

∞ X k 2 akX = a0 + a1X + a2X + ··· , ai ∈ R, k=0 and all such formal power series together form the ring R[[X]] of formal P k power series with coefficients in R. (Multiplication is defined by k akX · P k P Pk k k bkX = k n=0 anbk−n X , as in the case of polynomials.) A formal power series is called a convergent power series if it has a positive radius −1/n of convergence r := limn→∞ inf |an| . The convergent power series together form the ring R{X} of convergent power series. (5) A formal Laurent series in the variable X with coefficients in k is an expression of the form

∞ X k n n+1 akX = anX + an+1X + ··· , ai ∈ k, n ∈ Z, k=n and all such formal Laurent series together form the field k((X)) of formal Laurent series with coefficients in k. A formal Laurent series is called a convergent Laurent series if it has a positive radius of convergence. The convergent Laurent series together form the field k({X}) of convergent Laurent series.

10 take quotient ﬁeld take algebraic closure ......

...... k[X] ...... k(X) ...... k(X) polynomials rational functions algebraic functions ...... 1/d k{X} ...... k({X}) ...... lim k({X }) −→ convergent power series convergent Laurent series convergent Puiseux series ...... 1/d k[[X]] ...... k((X)) ...... lim k((X )) −→ formal power series formal Laurent series formal Puiseux series

Figure 2.1: Several structures of functions together with their relations.

(6) A formal Puiseux series in the variable X with coeﬃcients in k is an expression of the form

∞ X k/d n/d (n+1)/d akX = anX + an+1X + ··· , ai ∈ k, d ∈ N, n ∈ Z, k=n and all such formal Puiseux series together form the field of formal Puiseux series with coefficients in k, which we shall denote by k((X)) or lim k((X1/d)) −→ in fancy algebraic notation. A formal Puiseux series is called a convergent Puiseux series if it has a positive radius of convergence. The convergent Puiseux series together form the field of convergent Puiseux series, which we shall denote by k({X}) or lim k({X1/d}). −→ (7) An analytic function, or holomorphic function, is a function from some domain U ⊂ C to C that equals in some neighborhood of each point P k z0 ∈ U some convergent power series k ak(z − z0) . For a fixed domain U ⊂ C, all such analytic functions U → C together form the ring of analytic functions on U. (8) A meromorphic function is a function f from some domain U ⊂ C to Ct{∞} for which there exists a discrete set of points V ⊂ U such that fU\V is an analytic function. For a fixed domain U ⊂ C, all such meromorphic functions U → C t {∞} together form the field M(U) of meromorphic functions on U.

Remark 2.2 (Generalization to more variables). If we take the polynomials in the variable X2 with coefficients in the ring R[X1], we get the polynomials in the variables X1 and X2 with coefficients in the ring R. In this way we can inductively define the ring of polynomials in the variables X1,...,Xn with coefficients in the ring R, which we shall denote by R[X1,...,Xn]. In the

11 same way we can generalize all other classes of functions from the previous example to a finite number of variables. Remark 2.3 (Identification of polynomials and functions). Strictly speaking, the polynomials cannot be identified with functions, because for some rings R several polynomials can induce the same function (the same holds for other types from Example 2.1). An element of R[X] induces a function R → R, and some of these functions are represented by elements of R[X]. For the infinite rings R we shall use there will always be a unique representative. Therefore we use the term polynomial both for an element of R[X] and its induced mapping R → R. Remark 2.4 (Shifting the variable). Sometimes we want to identify for instance k{X} with the convergent power series functions in a point a ∈ k. To make this identification then more explicit, we shall write k{X − a} for k{X}. Changing the symbol does not change the mathematics, so this is just some convenient notation. The following theorem gives us the relations between the several structures of functions from Example 2.1, and similar results hold when there are more variables. For a schematic representation of these relations, see Figure 2.1. Theorem 2.5. We have the following relations between the structures of functions from Example 2.1. 1. The field of fractions of k[X] is k(X).

2. The ﬁeld of fractions of k{X} is k({X}).

3. The ﬁeld of fractions of k[[X]] is k((X)).

4. The algebraic closure of k(X) is k(X).

5. The algebraic closure of k({X}) is lim k({X1/d}). −→ 6. The algebraic closure of k((X)) is lim k((X1/d)). −→ Proof. The ﬁrst four statements are by deﬁnition or immediate. For a proof of the statements (5) and (6), which constitutes the Puiseux Theorem, see [25, Theorem 3.1] for an elementary and constructive proof. Van der Waer- den gives a shorter proof in [24, II, par. 14] that is due to Ostrowski.

2.2 Factorizing a Polynomial Diﬀerential Equation

In the previous section we discussed several structures of functions and the connection between these structures. In this section we shall use this to ﬁnd the factorization of a polynomial diﬀerential equation. We shall use the following immediate consequence of the Puiseux Theorem:

12 Corollary 2.6 (Factorization of a polynomial equation). Suppose we have a polynomial

m F (S, T ) = P0(T )S + ··· + Pm(T ),P0(T ) 6≡ 0,F ∈ C[S, T ].

For a ﬁxed a ∈ C, there exists a unique set {S1(T ),...,Sm(T )} ⊂ C(((T − a)1/d)) such that

m Y F (S, T ) = P0(T ) (S − Si(T )) . i=1

The form of a Puiseux series Si(T ) from the previous corollary depends on P0(T ) and on the S-discriminant ∆(T ), which are both polynomials in T . We shall need the following theorem in Chapter Classical Painlev´eTheory.

Theorem 2.7. In the setting of the previous corollary, with F ∈ C[S, T ] 1/d irreducible, we have that the Si(T ) ∈ C({(T − a) }). Furthermore, we have the following.

• If P0(a) 6= 0 and ∆(a) 6= 0, then the Si(T ) have no terms with negative exponents and no terms with nonintegral exponents. That is, the Si are power series in T .

• If P0(a) 6= 0 and ∆(a) = 0, then the Si(T ) have no terms with negative 1/d exponents. That is, the Si(T ) are power series in T for some d.

• If P0(a) = 0, then there exists a Si(T ) starting at a term with negative exponent.

Proof. For every but the last claim, see for instance [24, Sections II.13- 14]. For the last claim, suppose that all Si(T ) have no nonzero terms with negative exponents. Then also sums and products of these Si(T ) have no nonzero terms with negative exponents, implying that the Pk(T )/P0(T ) have no nonzero terms with negative exponents. This contradicts the irreducibility of F , so we conclude that such a Si(T ) must exist.

2.3 Local Rings and Valuations

In Chapter Modern PainlevéTheory the tools for studying differential equations locally will be local rings and valuations. This section briefly discusses these notions together with some additional terminology. In what follows assume that K ⊃ C is a field extension (later we take this to be a differential field extension).

Deﬁnition 2.8. A valuation ring over C of a ﬁeld K ⊃ C is a ring O such that:

13 V1: C ( O ( K; V2: a ∈ K\O =⇒ a−1 ∈ O.

One can show that a valuation ring O is a local ring. That is, it has a unique maximal ideal. These maximal ideals play an important role, and therefore have their own name.

Deﬁnition 2.9. A maximal ideal of a valuation ring of K over C is called a place over C of K.

Remark 2.10. The terminology place comes from the fact that the places of a function field correspond to the points on the corresponding curves. Remark 2.11. A place determines its corresponding local ring. If p is a −1 place of K ⊃ C, then its local ring is given by Op := {z ∈ K | z ∈/ p}. Furthermore, places in an algebraic function field correspond one-to-one to the points on the corresponding algebraic curve. When an element f of the function field K is not an element of this local ring Op, we say that p is a pole of f.

Definition 2.12. Let p be a place in the field K ⊃ C. The field Kp := Op/p is called the residue (class) field of p. The degree of a place p is defined as deg p := [Kp : C]. Places of degree one are called C-rational points. The following lemma is also known as the valuative criterion of properness. It is a duality between places and discrete valuation rings, and we shall need it in Chapter Modern PainlevéTheory. For a proof, see for instance [7, Chapter 7, Theorem 1, Corollary 4] or [9, Chapter 1, Corollary 6.6].

Lemma 2.13. Let X be a nonsingular projective curve and K ⊃ C its function ﬁeld. Then there is a one-to-one correspondence between the points of X and the discrete valuation rings of K ⊃ C. If x ∈ X, then Ox = OX,x is the corresponding discrete valuation ring.

We shall use the following lemma over and over again. It tells us that we can identify the local ring of a nonsingular point with some subring of the convergent power series. This will enable us to make explicit calculations in local rings.

Lemma 2.14. Let Ox be a local ring corresponding to a nonsingular point x of a curve X, and let π be a local parameter at x. The mapping

∞ X n Ox −→ C{π}, f 7−→ anπ , n=0

Pk n k+1 where the an are deﬁned by f − n=0 anπ ∈ (π ) for all k ∈ Z≥0, is an inclusion.

14 Proof. See for instance [20, Chapter II.2.2, Theorem 5].

If we take the field of fractions of a local ring Op, then we obtain the −1 function field. In other words, if z∈ / Op then z ∈ Op. Once we have such a power series representation of the elements of Op, we can therefore extend it to a Laurent series representation of the whole function field. More precisely, we have the following corollary.

Corollary 2.15. Let z0 be a point in C that is unramified (see below) in some finite field extension C ⊃ C(z). Then there exists an embedding C,→ C({z − z0}).

2.4 Ramiﬁcation and Branch Points

In this section we shall talk about the local structure of dense morphisms between curves, and in particular redeﬁne branch points. The following theorem makes this local structure explicit.

Theorem 2.16. For every dense morphism f : X → Y of nonsingular curves over C, and for every x ∈ X, there exist neighborhoods U 3 x and V 3 f(x) and homeomorphisms u : U → C and v : V → C onto neighborhoods of 0 in C such that the diagram

u ... U ...... C ...... f . . ρk ...... V ...... v ...... C k is commutative. Here ρk(z) = z , where k is deﬁned as the order of the zero of the function f ∗(t) at x, for t a local parameter at f(x).

Proof. See [21, Chapter VII.3.1, p. 131-3].

This theorem shows us that every dense morphism locally looks like the k map z 7→ z , for a certain k ∈ N. This leads to the following definition. Definition 2.17. The number k from the previous theorem is called the ramification degree of f at x ∈ X. If for some x ∈ f −1(y) the ramification degree of f at x is greater than 1, then y is called a branch point or ramification point of f.

15 2.5 Diﬀerential Function Fields

In the classical theory of differential equations, the basic object was always an equation of the form F (z, y0, y) = 0. Furthermore, solutions y were always real or complex-valued functions defined on some domain of R or C. In this section we shall define a new object, called a differential function field, that will in Chapter Modern PainlevéTheory replace the classical notion of a differential equation. In the following let A be a ring and let B be an A-algebra, both commutative, with unity and without zerodivisors. Definition 2.18. A mapping d : B → B is called an A-derivation if it satisfies the following conditions for all x ∈ A and all y, z ∈ B: D1: d(y + z) = d(y) + d(z);

D2: d(yz) = yd(z) + d(y)z;

D3: d(x) = 0. It is also called an A-derivation on B. In many cases we shall denote the derivation of an element x simply by x0. Proposition 2.19. Under the assumption that B has no zerodivisors, we can extend a derivation d : B → B to a derivation on the quotient ﬁeld K of B in one and only one way.

Proof. Uniqueness: Let db : K → K be an arbitrary derivation on K equal to the derivation d on B. Then for each x ∈ B we have that 1 1 1 1 = db(x · ) = xdb( ) + d(x) x x x implying that for all x, y ∈ B, y 6= 0, we have x 1 x db( ) = d(x) − d(y). (2.1) y y y2

The derivation db is therefore determined once it is defined on B. Existence: If we define db: K → K by formula 2.1, then it is a straightforward calculation to show that the axiom’s D1 and D2 of a derivation are satisfied.

Several notions from field theory generalize directly to notions with an additional derivation structure. More precisely, we have the following definitions. Definition 2.20. A differential ring is a pair (R, d), where R is a ring and d : R → R is a derivation. Sometimes we shall denote the differential field by R for short. If R is actually a field, then we call R a differential field.

16 Definition 2.21. An element of a differential field whose derivative vanishes is called a constant. The set of all constants in a differential field is called the field of constants. It is straightforward to show that the field of constants is indeed a field. It is a trivial example of a differential subfield. Definition 2.22. A subfield K ⊂ L of a differential field is called a differential subfield of L if it is closed under the differentiation on L. The field extension L ⊃ K together with an extension of the deriva is called a differential field extension of K. Definition 2.23. An extension field K ⊃ C is called an algebraic function field of one variable over C if there is an element x ∈ K that is transcendental over C, and K is algebraic of finite degree over C(x). It will be called a differential (algebraic) function field of one variable over C if C is a differential subfield of K. With these new notions of differential rings and differential fields come new questions about how morphisms between them behave, in particular under extensions of the fields.

Lemma 2.24. Suppose φ : R1 ,→ R2 is an inclusion of differential rings, where R1 and R2 are commutative and have no zerodivisors. Then φ can be extended uniquely to an inclusion φb : F1 ,→ F2 of differential fields, where Fi is the field of fractions of Ri.

Proof. Uniqueness: Suppose we have a homomorphism φb : F1 → F2 of differential fields that restricts to φ on R1. Then, for any x ∈ R1, we have 1 1 that 1 = φ(x · x ) = φ(x) · φb( x ) implies that φb is uniquely determined by the formula x x φ( ) = φ(x) · φ(y)−1, for all ∈ F (2.2) y y 1 and thereby by its definition on R1. Existence: Define φ : F1 → F2 by equation 2.2. This can be done since φ(y) = 0 implies that y = 0. It is straightforward to show that this is an inclusion of differential fields that restricts to φ on R1. Lemma 2.25. Let d : K → L be a derivation, and suppose that 0 6= x ∈ L is algebraic over K. Then d can be extended to a derivation db : K(x) → L in one and only one way.

Proof. Uniqueness: Assume we have such a derivation db, and let f(X) = Pn k k=0 akX ∈ K[X] be the minimal polynomial of x. Then n n X k X k−1 0 = d(f(x)) = d(ak)x + kakx db(x), k=0 k=1

17 and consequently

Pn k k=0 d(ak)x db(x) = −Pn k−1 , (2.3) k=1 kakx which is well-defined because every minimal polynomial is separable when char(K) = 0. It follows that the derivation db is determined by d. Existence: Define db: K(x) → L by formula 2.3. Then it is a straightforward calculation to show that db equals d on K and that it satisfies D1 and D2.

Theorem 2.26 states an analogous result for a transcendental extension. The following lemma shows how an algebraic function field of one variable over a given differential field C becomes a differential extension of C.

Theorem 2.26. Let K be an algebraic function field of one variable over a given differential field C. Let x be a separating variable in K. For any element y ∈ K there is a unique derivation db : K → K such that db(x) = y and it coincides with the given derivation d : C → C.

P i Proof. If φ is an element of C[x] of the form aix , then we deﬁne

X i X i−1 db(φ) = d(ai)x + y iaix .

Clearly it is the unique derivation of C[x] into K that satisﬁes

db(x) = y, db(a) = d(a), for all a ∈ C.

By the arguments of Proposition 2.19, dbcan be extended to C(x) in a unique way. Since K is separably algebraic over C(x), Lemma 2.25 implies that db can be extended in a unique way to a derivation on K.

18 CHAPTER 3 Classical Painlev´eTheory

n this thesis we examine PainlevéTheory from two points of view: the I classical function-theoretic approach and a modern differential algebro- geometric approach. First of all we shall in this chapter consider Painlevé Theory from the classical point of view. In Section 3.1, we shall give an overview of several primary and secondary sources in the classical literature concerning first order equations with the PP. After this, we shall discuss two of the most important theorems appearing in the primary sources, starting in Section 3.2 with a theorem of Briot and Bouquet for autonomous equations. We shall restate this theorem and its proof in a more readable way, and show that it generalizes the result we found in Chapter Introduction for equations of the form y0p = yq. In Section 3.3 we shall turn this theorem into an algorithm that decides if an autonomous first order equation has the PP. Finally, in Section 3.4, we shall discuss the generalization of the theorem of Briot and Bouquet to general first order equations, called Fuchs’s Criterion.

3.1 First Order Painlev´eTheory in the Literature

This section is concerned with the appearance of first order PainlevéTheory in the literature. It is often hard to say when a historical development begins, and the same holds for PainlevéTheory. One could argue that it started at the moment that multivalued functions were understood better, functions that strictly speaking do not live on the complex plane but on some covering space. This was at the time of Puiseux. Victor Alexandre Puiseux was the first to make the distinction between poles, essential singularities and branch points [2, p. 571]. Furthermore, he had made a theorem that can be stated nowadays as that the algebraic

19 (a) The French mathemati- (b) The French mathematician Victor Alexandre Puiseux cian Jean Claude Bouquet (1820-1883). (1819-1885).

Figure 3.1: Two 19th century mathematicians who layed the foundations for PainlevéTheory. closure of the field of (convergent) Laurent series is the field of (convergent) Puiseux series [19]. His work on these functions cleared the road for a better understanding of multivalued functions and thereby a better understanding of multivalued solutions of differential equations. Briot and Bouquet, who knew Puiseux from the Ecole´ Normale Supérieure, studied autonomous differential equations of the form F (y0, y) = 0, where F is a rational function in y0 and y. They asked themselves when such equations have a solution with a branched point. In 1856 they published three important articles in the Journal de l’Ecole´ (Imperiale) Polytechnique. The first one contained some notions from the theory of complex variables, the second was about differential equations in the complex domain and the third about the integration of such differential equations by means of elliptic functions. In this last article, l’Intégration des équationsdifférentielles au moyen des fonctions elliptiques [3], they answered the question when such a differential equation has a solution with a branch point. More precisely, they state and prove one way (the necessity of the conditions) of the following theorem and remark that the other way is trivial.

Theorem 3.1 (Briot and Bouquet). Pour qu’une équation différentielle du premier ordre de la forme

dum dum−1 + f (u) ··· + f (u) = 0 dz 1 dz m

20 ◦ admette une intégrale monodrome: 1 les coefficients f1(u), f2(u), . . . , fm(u) doivent être des polynômesentiers en u et, au plus, le premier du second degré,le second du quatrièmedegré,.. . ,le dernier du degré 2m; 2◦ quand, pour une certaine valeur de u, l’équationa une racine multiple différente du ◦ de zéro, dz doit rester monodrome par rapport à u; 3 quand, pour une certaine valeur u1 de u, l’équation a une racine multiple égaleàzéro, le du premier terme du développement de dz , suivant les puissances croissantes 1 n n−1 de (u − u1) , doit avoir l’exposant n , si cet exposant est plus petit que l’unité; 4◦ enfin l’équationdifférentielle que l’on déduit de la première en 1 posant u = v , doit offrir, pour v = 0, les mêmescaractères. They then used this theorem to carry out a detailed analysis of the du m binomial equations dz = f(u), with f a polynomial in u. In Section 3.2 we shall rephrase this theorem in a more readable form and prove it in a classical way. As we can see, the criterion is technical and does not (directly) give us much insight. It is, however, correct, and in Section 3.3 we turn this theorem into an algorithm to determine if such an autonomous first order equation has the PP. Something else that was already found by Briot and Bouquet is the classification of all autonomous equations with the PP by the genus of the associated curve, into the Riccati and the Weierstrass equations. Altogether we can say that Briot and Bouquet completely solved the autonomous case, thereby laying the foundation for the general case. The next person who made a contribution was Lazarus Immanuel Fuchs (who should not be confused with his less famous son R. Fuchs, who con- tributed to the second order equations with the PP). In 1884, Fuchs tried to generalize Briot and Bouquets theorem to nonautonomous differential equations but made some mistakes [6]. More precisely, he makes the following claim.

Claim 3.2 (Fuchs). Die nothwendigen und hinreichenden Bedingungen daf¨ur, dass die Integrale der Gleichung F (z, y0, y) = 0 feste, sich nicht mit den Anderungen¨ der Anfangswerthe stetig verschiebende Verzweigungspunkte besitzen, sind die folgenden:

1. Die Gleichung hat die Form:

0m 0m−1 0m−2 y + ψ1y + ψ2y + ··· + ψm = 0,

worin ψ1, ψ2, . . . , ψm ganze rationale Functionen von y mit von z ab- hängigenCoefficienten von der Beschaffenheit bedeuten, dass ψk höch- stens vom Grade 2k in Bezug auf y ist.

2. Ist y = η eine Wurzel der Discriminantengleichung (C.), f¨urwelche die durch (F.) deﬁnirte algebraische Function y0 von y sich verzweigt,

21 (a) The French mathemati- (b) The German mathe- cian Henri Poincar´e (1854- matician Lazarus Immanuel 1912). Fuchs (1833-1902).

Figure 3.2: Two mathematicians who solved the general case for ﬁrst order equations.

so ist η ein Integral der Gleichung (F.). In der y0 als algebraische Func- tion von y darstellenden Riemannschen Flächehat y0 in sämtlichen 0 dη über y = η liegenden Verzweigungsstellen den Werth y = ζ = dz . 0 dη 3. Je α Blättern, welche sich in y = η, y = ζ = dz verzweigen, entsprechen mindestens α − 1 mit y = η zusammenfallende Wurzeln der Gleichung

F (z, y, ζ)

mit der Unbekannten y. This criterion determines whether a first order differential equation of the form F (z, y0, y) = 0, F rational in y0 and y, has the PP. It is strange that he does not specify its dependence on z, because it is in this that he generalizes the theorem of Briot and Bouquet. One could for instance take F to be algebraic in z. Although all the conditions in the criterion are necessary, they are altogether not sufficient. It is strange that Fuchs made this mistake, because it is easy to think of examples of functions that do pass Fuchs’s test but do not pass the test of Briot and Bouquet. As we shall see in Section 3.4 the criterion can be made sufficient by adding an extra condition, thus obtaining what is nowadays known as Fuchs’s Criterion. Poincaré’s response to Fuchs’s article was an article published in 1885 in which he considered the case of a genus bigger than two [18]. After this, Paul Painlevépublished an article in 1885 [16] on the first order case, combined all previously obtained results together with his new results for the second

22 order equations into his Stockholm Lectures [17] (that appeared in 1895), thereby creating PainlevéTheory. But PainlevéTheory is a tricky business, because once again many mistakes (at least for the second order case) were made in this work. The classification of the second order equations into the six Painlevéequations known nowadays was completed by his student B. Gambier, but the proof that all these equations do have the PP themselves was done only a few years ago by Masahiko Saito et al. at Kobe University. So the PainlevéTheory constructed so far was far from perfect. In 1927 the British mathematician Edward Lindsay Ince wrote a book in which he summed up the most important results and techniques developed so far for ordinary differential equations in a book called Ordinary Differential Equa- tions [11]. Although the book became a classic and a standard reference, its section about first order PainlevéTheory is incomplete, imprecise and sometimes wrong. Despite these shortcomings, many mathematicians take this book to be the standard reference concerning first order PainlevéTheory. In 1909, the Swedish mathematician Axel Johannes Malmquist (1882 - 1952) wrote his dissertation called Sur les équationsdifférentielles du premier ordre dont l’integrale generale admet un nombre fini de branches per- mutable autor des points critiques mobiles under the supervision of Gösta Mittag-Leffler. This dissertation was followed by an article in 1913 on the same topic [13], and then no further work was published for many years. In 1940, he all of a sudden started publishing results again. He concluded this publishing spree with an article in which he gives the classification of the first order equations with the PP in a very precise form, but the proofs are very classical and therefore very difficult to understand [14]. In a next attempt to sum up what exactly had been done in the area of first order equations with the PP, the German mathematician and Nazi Ludwig Georg Elias Moses (!) Bieberbach wrote a book in 1953, titled Theorie der gewöhnlichen Differentialgleichungen: auf funktionentheoretische Grundlage dargestellt, in which he recited several classical statements and proved some of them [1]. These statements and proofs are much more precise than what was done in Ince’s book. In 1976, the Swedish mathematician Einar Carl Hille wrote the book Or- dinary Differential Equations in the Complex Domain [10] simply because, as he stated in the preamble, there was at that moment not yet a book dedi- cated solely to this topic. In quite a large part of the book he considers first order equations with the PP and proves in a very precise and readable way results about the explicit differential equations y0 = R(z, y), R a rational function of z and y.

23 (a) The German mathemati- (b) The Swedish mathematician and Nazi Ludwig Georg cian Einar Carl Hille (1895- Elias Moses Bieberbach (1886- 1980). 1982).

Figure 3.3: Two mathematicians who redid some classical Painlev´eTheory.

3.2 Classical Proof of the Theorem of Briot and Bouquet

In this section we shall restate and reprove the theorem of Briot and Bouquet that we encountered in Section 3.1. We start with some remarks that might help in the understanding of the proof of the theorem. Remark 3.3. In what follows, the reader should be careful and realize that P k/d we work with two types of Puiseux series: those of the form k aky that P k/e help to factorize the differential equation and those of the form k akz that denote solutions of the differential equation. Remark 3.4. We first remark that the equation not having a solution with a movable branch point is equivalent to the equation not having a solution with a branch point at 0. To see this, suppose that the equation has a solution w with a branch point at 0. Then, since the differential equation is autonomous, the function Φ:(z, t) 7−→ w(z + c(t)) has the PP for a sufficiently small neighborhood U of zero and every nonconstant path c : [0, 1] → U with c(0) = 0. On the other hand, if a differential equation has a solution with a movable branch point then it has of course a solution with a branch point at 0 as well. Within the proof of the theorem of Briot and Bouquet, we make use of the following lemmas several times.

24 Lemma 3.5. Suppose we have a function w : U → C, where U is a neighborhood of z0, that can be expressed as a convergent power series of the form ∞ X k w − w0 = wk(z − z0) , w1 6= 0. k=1

Then the function w − w0 has an inverse z − z0, deﬁned in a neighborhood of 0, that can be expressed as a convergent power series as well, given by ∞ 1 X k z − z0 = (w − w0) + zk(w − w0) . w1 k=2 Proof. It follows from the complex version of the implicit function theorem that there exists locally an inverse that can be expressed as a convergent power series. Composing power series shows that it must be of the given form.

Lemma 3.6. Suppose we have a function w : U → C, where U is a neighborhood of z0, that can be expressed as a convergent Puiseux series of the form ∞ X k/d w − w0 = wk(z − z0) , wn 6= 0, n ≥ 1, d ≥ 1. k=n

Then the function w − w0 has an inverse z − z0, deﬁned in a neighborhood of 0, that can be expressed as a convergent Puiseux series of the form

" ∞ #d X k/n z − z0 = zk(w − w0) , z1 6= 0. k=1 Proof. Because we can write every power series with nonzero constant term as an arbitrary power of some other power series with nonzero constant term, we ﬁnd that

" ∞ #n n/d X 0 k/d 0 w − w0 = (z − z0) wk(z − z0) , w0 6= 0, n ≥ 1, k=0 for a certain convergent Puiseux series between the brackets. Taking on each −1 1/n 1/d side the n th powers, we ﬁnd that (w − w0) as a function of (z − z0) has (by Lemma 3.5) an inverse whose dth power is of the form proposed in the lemma.

Lemma 3.7. Suppose w : U → V and its inverse (under composition) z : V → U are functions on certain domains U, V ⊂ P that have expansions dX as a Puiseux series in their whole domain. If w is a solution to dt = F (X) such that F (w(z)) 6= 0 for every z ∈ U, then its inverse (under composition) dX 1 z is a solution to dt = F (t) .

25 Proof. Using the chain rule in the differential field of convergent Puiseux series, we find that d(Id ) d(w ◦ z) dw dz dz 1 ≡ V (t) = (t) = (z(t)) · = F (t) · , dt dt dz dt dt from which the lemma follows since F (t) 6= 0 for every t ∈ V .

Now we are prepared to state and prove the theorem of Briot and Bou- quet. Theorem 3.8 (Briot and Bouquet, 1856). Consider the diﬀerential equation

0 0m F (y , y) = P0(y)y + ··· + Pm(y) = 0,F ∈ C[S, T ] (3.1) irreducible with S-discriminant ∆(T ), P0 6≡ 0. For every a ∈ C, we can write F (S, T ) in a unique way in the form

m Y 1/d F (S, T ) = P0(T ) (S − Si(T )) ,Si ∈ C({(T − a) }). i=1 The diﬀerential equation has no solution with a movable branch point if and only if the following criteria for F hold:

A1: For k = 0, . . . , m we have that deg Pk ≤ 2k.

A2: If, for a certain a ∈ C, ∆(a) = 0 and one of the Si(T ) has a nonzero constant term, then this Si(T ) has only terms with integral exponents.

A3: If, for a certain a ∈ C, ∆(a) = 0 and one of the Si(T ) has a zero constant term and starts at a term with an exponent less than one, (d−1)/d then this Si(T ) starts at a term of the form (T − a) .

0 0 0 2 A4: If we transform the equation F (y , y) = 0 by y = 1/yb and y = −yb /yb , then we obtain an equation of the form

0 0m Fb(yb , yb) = Pb0(yb)yb + ··· + Pbm(yb) = 0, Fb ∈ C[S, T ] (3.2)

irreducible with S-discriminant ∆(b T ), Pb0 6≡ 0. For every a ∈ C, we can write Fb(S, T ) in a unique way in the form

m Y 1/d Fb(S, T ) = Pb0,a(T − a) S − Sbi(T ) , Sbi ∈ C({(T − a) }). i=1 The criteria A2 and A3 hold for a = 0, if we omit the wide hats. The following proof is a mod(ern)iﬁcation of the original proof by Briot and Bouquet, given in their article [3].

26 Proof. Necessity. Let us ﬁrst prove that the conditions A1-A4 are necessary. Assume that the equation does not have a solution w with a branch point at 0. We now prove one by one that the criteria are necessary. A1. Suppose deg P0 > 0. According to the fundamental theorem of algebra, there exists an a ∈ C such that P0(a) = 0. By Theorem 2.7, there 1/d exists a Puiseux series Si(T ) ∈ C({(T − a) }) starting at a nonzero term −1 with negative exponent. It follows that Si(T ) starts at a nonzero term with positive exponent. That is, it is a convergent power series in the symbol 1/d dz −1 (T −a) . Let z(T ) be a solution of the diﬀerential equation dT = Si(T ) , that can be expressed as a convergent Puiseux series of the form

∞ X i/d z = ci(T − a) , cn 6= 0, n > d ≥ 1. i=n Then, by Lemma 3.6, z(T ) has an inverse (under composition) w(z) that can be expressed as a Puiseux series of the form

" ∞ #d X i/n w − a = wiz , w1 6= 0. i=1 According to Lemma 3.7, w(z) is a solution of the differential equation dw dz = Si(w). Since n > d, the Puiseux series expansion of w(z) starts at a term with a rational exponent, and it follows that this solution w(z) has a branch point. If there ought to be no solutions with branch points, then it follows that deg P0 = 0. Now suppose deg Pl > 2l for a certain 1 ≤ l ≤ m. Performing the transformation in A4, we obtain a new differential equation of the form of (3.2). It now follows from the fact that Fb is a polynomial that Pb0(yb) has a factor yb. By the previous paragraph, this implies that there exists a solution v to (3.2) with initial condition v(0) = 0 and a branch point at 0. This means, however, that there exists a solution w = 1/v to (3.1) with initial condition w(0) = ∞ and a branch point at 0. We conclude that if there ought to be no solutions with branch points, it follows that deg Pk ≤ 2k for every k = 0, . . . , m. In what follows assume that A1 holds and in particular that P0(a) 6= 0 for every a. By Theorem 2.7 it then follows that the Si(T ) have no terms with negative exponents. A2. Assume that for a certain a ∈ C one has ∆(a) = 0 and one of −1 the Sl(T ) starts at a nonzero constant term. Then also Sl(T ) starts at dz a nonzero constant term. It follows that the differential equation dT = −1 Sl(T ) has a solution z(T ) that can be expressed as a convergent Puiseux series starting at a nonzero term with exponent one. By Lemma 3.6, its inverse (under composition) w(z) exists and can be expressed as a Puiseux series in z. According to Lemma 3.7, w(z) is a solution to the differential

27 dw equation dz = Sl(w), with initial condition w(0) = a. Since this equation has no solutions with branch points, w and w0 must necessarily be of the form ∞ ∞ X i 0 X i w − a = w1z + wiz , w = w1 + (i + 1)wi+1z , w1 6= 0. i=2 i=1 Using Lemma 3.5, it follows from the form of the Puiseux series of w(z) that the inverse (under composition) z(w) can be expressed as a convergent Puiseux series of the form ∞ 1 X z = (w − a) + z (w − a)i. w i 1 i=2

Substituting this in the expression for w0, we ﬁnd that the Puiseux series of w0 in w − a necessarily has no nonzero radical terms if there ought to be no solutions with branch points. Subtracting this Puiseux series from Sl(w), we ﬁnd that Sl(w) must be of the same form. A3. Assume that for a certain a ∈ C, one has ∆(a) = 0 and one of the 1/d Sl(T ) ∈ C({(T − a) }) is of the form

X k/d Sl(T ) = ck(T − a) , cn 6= 0, 0 < n < d ≥ 2. k=n

dz −1 Then the equation dT = Sl(T ) has a solution z(T ) that starts at a term with exponent (d − n)/d. Its inverse (under composition) w(z) exists by Lemma 3.6 and starts at a term with exponent d/(d − n). According to dw Lemma 3.7, it is a solution of the equation dz = Sl(w). Since this equation is not allowed to have solutions with branch points, it follows that d−n = 1, and we find ∞ ∞ X k X k/d w − a = wkz , z = zk(w − a) , wd, z1 6= 0, d ≥ 2. k=d k=1 Differentiating the expression for w − a and substituting the expression for z we find that ∞ ∞ 0 X k X k/d w = (k + 1)wk+1z = bk(w − a) , bd−1 6= 0, k=d−1 k=d−1 showing that under the assumptions of criterion A3, it must necessarily follow that the Puiseux series of w0 in w − a starts at the nonzero term with exponent (d − 1)/d, if there ought to be no solutions with branch points. Subtracting this Puiseux series from Sl(w), we find that Sl(w) must be of the same form.

28 A4. This criterion says that the criteria A2 and A3 must hold for the initial condition w(0) = a not considered so far; namely a = ∞. This is the only point of the Riemann sphere that does not occur in the chart chosen so far, and we must check this case separately by switching to a chart around ∞. If in this new chart one of the criteria fails at infinity, then it follows by definition and by the just proved necessity of the criteria A2 and A3 that there is a solution v of the transformed equation (3.2) that has a branch point somewhere. Then there also exists a solution w = 1/v of the original (3.2) equation that has a branch point. This is a contradiction, and we conclude that for a = 0 the criteria A2 and A3 must hold for the transformed equation. Sufficiency. Assume the differential equation satisfies the properties A1-A4. It follows from criterion A1 that there does not exist an a ∈ C such that P0(a) = 0, and it follows from the criteria A1 and A4 that the same holds at infinity. What remains is the case that P0 is a nonzero constant polynomial. We prove that the criteria are sufficient, by distinguishing three possible cases and using in each case some of the criteria to conclude that there can be no solutions with branch points. In what follows, we shall use that a function w(z) is, in a certain neighborhood of 0, a solution to F (y0, y) = 0 with initial condition w(0) = a if and only if w satisfies one of the equations 0 y = Si(y) in a certain neighborhood of 0. Case I: a ∈ C and ∆(a) 6= 0. In this case the equation F (S, a) = 0 has 0 0 m different solutions w1, . . . , wm for S. It follows from Theorem 2.7 that the Si(T ) are all pairwise different and of the form

∞ 0 X k Si(T ) = wi + ak(T − a) , i = 1, . . . , m. (3.3) k=1

It is not difficult to show that when an Si(T ) starts at a term with exponent 0, 1 or 2, the corresponding solutions do not have branch points. On the 0 other hand, k ≥ 3 would, using that the wi are different, contradict A1. It follows that in this case the equation has no solutions with branch points. Case II: a ∈ C and ∆(a) = 0. In this case it follows by Theorem 2.7 from P0(a) 6= 0 that there exist l < m different Puiseux expansions of the form ∞ X k/d Si(T ) = ak,i(T − a) , aAi,i 6= 0, i = 1, . . . , l. (3.4) k=Ai≥0

We distinguish three cases, starting with the case that Ai ≥ d. Suppose that we have a solution w(z) = a + bzλ + ··· . Substituting this solution in 0 the equation w = Si(w) we ﬁnd that λ − 1 = λAi/d, which has no solutions for positive λ. Since w(0) 6= ∞, we conclude that this case does not occur.

29 Secondly, suppose that Ai = 0. It then follows from criterion A2 that Si(T ) is a power series in the symbol T − a, starting at a nonzero constant term. From this it follows directly that the solutions of the equation w0 = Si(w) are power series and, in particular, free of branch points. Thirdly, suppose that 0 < Ai < d. In this case it follows from criterion A3 that Si(T ) is of the form (d−1)/d d/d Si(T ) = b0(T − a) + b1(T − a) + ··· , b0 6= 0, 0 and suppose we have a solution w to the equation w = Si(w). Since w can 1/d be expressed as a Puiseux series in z, so can an dth root wb = w . This wb satisfies the equation b b b w0 = 0 + 1 w + 2 w2 + ··· , b 6= 0. b d d b d b 0 Since the r.h.s. is a power series expression in wb starting at a nonzero constant term, the Puiseux series wb is in fact a power series. Therefore d w = wb is itself a power series and is free of branch points. Case III: a = ∞. Due to criterion A4 this case reduces to one of the cases above. Let us check if the previous theorem confirms our result for differential equations of the type y0p = yq. We want to prove the following. Proposition 3.9. In the setting of Theorem 3.8, suppose that F (S, T ) = p q S − T , with p, q ∈ N, pq > 1 and gcd(p, q) = 1. Then the statement that |p − q| = 1 is equivalent to the statement that A1, A2, A3 and A4 hold. Proof. ‘=⇒’: Assume |p − q| = 1. Then A1 follows immediately, because deg P0 = 0 ≤ 2 · 0 and deg Pp = q ≤ p + 1 ≤ 2p. If ∆(a) = 0, then a = 0 and √ i q/p 2π −1/p Si(T ) = ζpT , ζp = e . This Puiseux series has a zero constant term, so the condition for A2 does not occur, and the criterion A2 is satisfied. The condition for A3 does occur only if q = p − 1, and A3 is satisfied in that case. If q = p ± 1 and one performs the transformation in criterion A4, then one ends up with a similar equation with q = p ∓ 1. It then follows by the same arguments that for a = 0 A2 and A3 hold for this new equation, thereby satisfying the criterion A4. ‘⇐=’: Assume A1, A2, A3 and A4 hold. If a = 0, then ∆(a) = 0 i q/p and we have again Si(T ) = ζpT . If q < p, then it follows from A3 that q = p − 1. On the other hand, if q > p and we apply the transformation of A4, then, because of A1, we end up with a similar polynomial Sp −T q0 with q0 < p. By A4, A3 must hold for this equation with a = 0. This then implies that q0 = p − 1. Transforming the equation back to the original equation, we then find that q = p + 1. Since p 6= q, we conclude that |p − q| = 1.

30 3.3 The Algorithm Indicated by the Theorem of Briot and Bouquet

In the previous section we stated and proved a theorem by Briot and Bouquet that gave necessary and suﬃcient conditions for an autonomous equation to have the PP. These conditions were very precise, and in this section we shall turn the theorem into an algorithm that determines whether or not an equation of the form

0 0m F (y , y) = P0(y)y + ··· + Pm(y) = 0,F ∈ C[S, T ] irreducible, has the PP. See algorithm 1 for the algorithm and appendix A for an implementation in Maple.

Algorithm 1. Determining if F (y, y0) = 0 has the PP ...... Require: An irreducible polynomial

m F (S, T ) = P0(T )S + ··· + Pm(T ) ∈ C[S, T ]. Ensure: A boolean containing the value true if the equation F (y0, y) = 0 has the PP and the value false if it has not. 1: for i = 0 to degS F do 2: if deg Pi > 2i then 3: return (false); 4: end if; 5: end for; 6: let G(s, t) := (denominator F (s/t2, 1/t)) · F (s/t2, 1/t); 7: if ∆s(0) = 0 then 8: let Lp be the set of triples (degree of branching, coefficient of constant term, order) for every Puiseux expansion of G in 0; 9: end if; 10: let Lc be the set of all branch points of F ; 11: for C ∈ Lc do 12: let Lp := Lp united with all triples (degree of branching, coefficient of constant term, order) for every Puiseux expansion of F in C; 13: end for; 14: for P ∈ Lp do 15: let n be the degree of branching of P ; 16: let z be the coefficient of the constant term of P ; 17: let o be the order of P ; 18: if z 6= 0 and n > 1 then 19: return (false); 20: end if; 21: if z = 0 and o < 1 and o 6= (n − 1)/n then

31 22: return (false); 23: end if; 24: end for; 25: return (true); ......

3.4 Fuchs’s Criterion

As we remarked in Section 3.1, Fuchs’s Criterion as stated in his article in 1884 is not sufficient. In this section we shall give some examples of equations that satisfy the original conditions of Fuchs but do not have the PainlevéProperty. After that we shall, inspired by a book written by the Japanese mathematician Michihiko Matsuda [15], give one extra condition that makes the criterion sufficient. We start with writing (the old) Fuchs’s Criterion in a more readable form, choosing algebraic functions as coefficients in the differential equation. To interpret Fuchs’s Criterion, we made use of a recent article written by the Chinese mathematicians Guoting Chen and Yujie Ma [4].

Theorem 3.10 (Fuchs’s Criterion, Classical). Let C ⊃ C(z) be a ﬁnite extension, and consider diﬀerential equations of the form

m 0 = F (S, T ) = A0S + ··· + Am−1S + Am,A0 6≡ 0,F ∈ C[S, T ] with S-discriminant ∆(T ) ∈ C[T ]. Then the following conditions are necessary to ensure that the diﬀerential equation F (y0, y) = 0 has the PP:

F1: The coeﬃcient A0 is independent of T . The equation may then be divided throughout by A0 and takes the form

0m 0m−1 0 y + ψ1(y)y + ··· + ψm−1(y)y + ψm(y) = 0,

in which the coeﬃcients ψk are algebraic in z and polynomials in y of degree at most 2k.

F2: Let y = η(z) be a root of ∆(y) = 0. If the equation

F (ζ, η(z)) = 0 (3.5)

has a multiple root ζ (or more of them), then η must be a solution of dη our original equation. That is, ζ = dz . dη F3: If the root ζ = dz of equation 3.5 has multiplicity α > 0, then y = η must be a root of F (ζ, y) of multiplicity equal to, or greater than, α−1. The following example shows that the conditions F1, F2 and F3 together are not suﬃcient.

32 Example 3.11. Consider the differential equation y03 = y5. According to Example 1.9, this equation does not have the PP since |3−5| = 1. Of course we could have used the theorem of Briot and Bouquet as well to find this. We are going to show that this equation does not have the PP but does satisfy F1, F2 and F3. The condition F1 is clearly satisfied, because 5 ≤ 2 · 3. To check conditions F2 and F3, we first find that the discriminant equals ∆ = −27p10, implying that the only root of ∆(z, η) ≡ 0 is η(z) ≡ 0. This then shows that the conditions F2 and F3 are satisfied. More generally we could have taken a differential equation 0p q x − x = 0, p, q ∈ N, gcd(p, q) = 1, 2p ≥ q > p + 1, and this example appeared in Matsuda’s book [15, Chapter 3, p. 15-6]. It is strange that Fuchs made this mistake if he had read the article of Briot and Bouquet that appeared thirty years earlier that formed the basis of the topic. A wild guess is that he had never seen the article because of the Franco-Prussian War that took place fourteen years before his publication. Matsuda correctly remarks that an additional condition is needed to make the criterion sufficient and provides the following condition.

F4: Suppose η has a pole at a ramification point p of X(C), then dη ord ( ) ≥ ord (η) − 1. p dz p Example 3.12. Consider again the differential equation F (y0, y) = y03 − y5 = 0,F ∈ C[S, T ]. We will show that this differential equation does not satisfy F4. Write the corresponding function field C(s, t) as C(u), with s = u5 and t = u3. Let p be the place corresponding to the point u = ∞. Then ordp(s) = −5 < −4 = ordp(t) − 1. Therefore the equation does not satisfy F4. Altogether we arrive at the following theorem.

Theorem 3.13 (Fuchs’s Criterion, Modern). Let C ⊃ C(z) be a finite extension, and consider differential equations of the form m 0 = F (S, T ) = A0S + ··· + Am−1S + Am,A0 6≡ 0,F ∈ C[S, T ] with S-discriminant ∆(T ). Then differential equation F (y0, y) = 0 has the PP if and only the conditions F1, F2, F3 and F4 hold. Proof. See [15, Chapter 3, p. 15]. Although Matsuda writes down this theorem in a fancy differential algebraic way, it is essentially the same.

33 CHAPTER 4 Modern Painlev´eTheory

n this chapter we consider PainlevéTheory from a modern point of view. I The advantage will be a more modern, differential algebro-geometric point of view in which the proofs are much easier to follow. All major proofs are due to Marius van der Put. The connection with algebraic geometry will relate the PP to the familiar notion of the genus of a curve, and the connection with differential algebra relates the PP to a crystal clear condition on the differentiation. Especially in this last condition, the derivation has no poles, we see how much clearer the modern theory is compared to the classical theory, which needed half a page for this (see Section 3.2). We start in Section 4.1 with describing the setting and theme of the chapter. In Section 4.2 we shall introduce the Algebraic PainlevéProperty, because this will be much easier to work with. The APP turns out to be very closely related to the PP, and in the remainder of the chapter we shall prove statements about this APP. In Sections 4.3 and 4.4 we shall prove that the Riccati equation and what we call the generalized Weierstrass equation have this APP. After that we shall, in Section 4.5, carry out the classification of the autonomous equations with the APP into Riccati equations and the Weierstrass equations and find our criterion on the associated function field equivalent to the APP. In Section 4.6 we shall do the same for the general case.

4.1 The Setting and Theme

In this section we shall explain the setting and theme used in the remainder of this chapter. The goal of this is to give the reader some feeling for what is done in this chapter. It is our hope that this will make it easier to grasp the meaning and signiﬁcance of the upcoming theorems.

34 In the classical theory of (complex) differential equations, the basic object was always an equation of the form F (z, y0, y) = 0, and solutions y were complex-valued functions defined on some domain of C. The function F was often taken to be some analytic or meromorphic function in y0 and y in 2 some neighborhood of C . In a general differential field K however, we cannot take every relation between z, y0 and y that we could take in the classical complex case. We need, for example, a notion of convergence in a function space to talk about a differential equation of the form F (z, y0, y) = 0 with F a power series in y0. This would take us back into the domain of analysis (though this time from a more modern point of view) thereby significantly complicating our situation. The best we can do in an arbitrary field is to apply a finite number of the operations +, −, · and / to its elements (when defined). In our general setting, our relation F can therefore be a rational function in y0 and y with elements from some differential subfield C of our differential field K as its coefficients. By multiplying with the common denominator, we can assume F to be an irreducible polynomial in y0 and y. Omitting its dependence on z in our notation, we write F = F (y0, y). We call the field C a coefficient field of the differential equation. As we shall see in Section 4.2, the statement that a differential equation has the APP is stable under finite extensions of its coefficient field. We can therefore assume that our polynomial F ∈ C[S, T ] is absolutely irreducible, that is, irreducible as an element of C[S, T ], because if F were not absolutely irreducible, we could take an appropriate finite extension of C without changing the fact of whether F (y0, y) = 0 has the APP or not. In this chapter we take C to be some finite extension of C(z), which is in some sense the simplest case. The proofs in this section will work as well for other coefficient fields, for instance for a finite extension of the field of meromorphic functions M(U) on some domain U of the Riemann sphere. Also direct limits of such fields of functions, for instance the field of convergent Puiseux series ({z − z }) = lim ({(z − z )1/m}), can be taken C 0 −→ C 0 for the coefficient field. As our differential field we take K = C(s, t), where t is transcendent over C, and s is algebraic over C(t). As we have seen, there is a unique lift of the derivation on C to a derivation on K up to the choice of d(t), which we can choose to be any element of K. We shall take the C-linear derivation defined by d(z) = 1, d(t) = s and the algebraic relation between s and t to be the polynomial F (s, t) = 0, where the coefficients are elements of C. That is, K is the fraction field of the coordinate ring C[S, T ]/(F ). In terms of this setting, a solution of the differential equation translates into a morphism of (some subring of) the differential function field K to one of the series encountered in Section 2.1. For an unramified local solution this morphism maps into the convergent Laurent series and for a ramified local solution into the convergent Puiseux series, and not into the Laurent series.

35 A global solution maps into the field M(C) of meromorphic functions on C. In the remainder of the chapter we assume that both S and T are present in the equation of F , thereby excluding some trivial cases. If S were not to appear in the equation of F , then our equation would technically not be a differential equation (there would not appear a derivative in the equation). Solutions of this equation are algebraic functions with branch points at fixed places. If on the other hand T were not to appear in the equation, then our differential equation would be of the form n 0 Y 0 0 = F (y ) = S0(z) (y − Si(z)), i=1 for some algebraic functions Ti(z). The solutions would then be integrals of algebraic functions, again with branch points at fixed places. Having said this, the problem we shall solve in this chapter will be the following. Problem 4.1. Let C ⊃ C(z) be a finite extension. Classify all differential equations with the APP of the form F (y0, y) = 0,F ∈ C[S, T ] absolutely irreducible, and both S and T present in the equation of F . We shall solve this problem completely and find that every such equation falls either in the category of the Riccati equation or in the category of the generalized Weierstrass equation. Therefore, we first prove that these equations have the APP and consequently prove the classification theorems in first the autonomous and second the general case. In the language of differential fields, we can restate this problem as follows. Problem 4.2. Let C ⊃ C(z) be a finite extension. Classify all differential extension fields (C(s, t), d) of transcendence degree one over C with the APP. Remark 4.3. Such a field C(s, t) corresponds to an absolutely irreducible nonsingular projective curve X given by an equation F (S, T ) = 0, unique up to isomorphism. It is a well known result that in every birational class of curves there exists exactly one nonsingular projective curve (see [20, Chapter II.3.1, Theorem 3, Corollary 2 and Chapter II.5.3, Theorem 7, Corollary 1]), which is called a nonsingular projective model. If we refer to the curve corresponding to the function field, then we shall always mean this curve (unless specified differently). Remark 4.4. In this formulation we can see directly which transformations are allowed to bring an equation in Riccati or Weierstrass form. When we have classified differential function fields up to isomorphism, then these correspond to the tranformations on the corresponding curves induced by the isomorphisms of function fields.

36 4.2 The Algebraic Painlev´eProperty

The classical definition of the PP was that when we vary the initial conditions slightly, the only problematic points that can vary in a continuous way with the initial conditions are poles. In this section we shall use a (on first sight) slightly more selective definition because it is much easier to work with.

Definition 4.5. We say that a differential equation has the Algebraic Painlevé Property (APP) , if there exists a finite number, possibly zero, of points z0 ∈ C for which the differential equation has a solution in C({z − z0}) that is not an element of C({z − z0}). A curve X given by an equation F (S, T ) = 0 has the APP if F (y0, y) = 0 has the APP, and a function field of a curve has the APP if the curve has the APP.

Remark 4.6. This definition means that there is only a finite number of points z0 on the Riemann sphere at which there exists a local branched solution. Obviously this implies that there can be no movable branch points, since that would require a curve in P on which lie uncountably many points where there exists locally a branched solution. On the other hand, suppose that there are infinitely many points z0 ∈ P for which there exists a branched solution in C({z − z0}). Since the Riemann sphere is compact (its topolo- 2 gical structure is isomorphic to S ), there exists a point of accumulation of these z0. This is as close as the definition gets to the PP. Remark 4.7. The APP is a stronger property than it seems to be on first sight. Although the APP is only concerned with branch points, we will see in the end of this chapter that equations with the APP also do not have movable essential singularities. Now we come to an important idea that will be omnipresent in the remainder of the chapter: the APP of a differential algebraic function field K ⊃ C is stable under finite extensions of the coefficient field C. That is, we can add a finite number of algebraic elements over C to K and C, without changing the fact of whether K ⊃ C has the APP or not. We make this more precise in the following lemma.

Lemma 4.8. Let K ⊃ C be a diﬀerential ﬁeld as in Section 4.1. For every x ∈ C, the statement “K ⊃ C has the APP” is equivalent to the statement “K(x) ⊃ C(x) has the APP.”

Proof. The inclusions of ﬁelds K ←-C, C(x) ←-C and K(x) ←-K induce morphisms X → Z, Z0 → Z and X0 → X of varieties. Since x is algebraic, the morphisms Z0 → Z and X0 → X can have only a ﬁnite number of branch points. We thus obtain a commutative diagram

37 0 ...... 0 X ... Z ...... XZ...... in which the vertical arrows denote morphism with only finitely many branch points. Therefore the morphism X → Z has finitely many branch points if and only if X0 → Z0 has finitely many branch points. This is what we had to prove.

4.3 The Riccati Equation

In the previous section we introduced the APP as a convenient (working) deﬁnition for the PP for studying them from a modern point of view. In this section we shall prove that one of the equations of our upcoming classiﬁcation has the APP. More precisely, we shall prove that the equation

T 0 A0(z) + A1(z)y + ··· + AT (z)y y = N ,Ai(z),Bi(z) ∈ C(z), B0(z) + B1(z)y + ··· + BN (z)y 0 has the APP if and only if it is a Riccati equation, y = A0(z) + A1(z)y + 2 A2(z)y . In order to use this result later in the chapter, we state it in terms of differential fields. The sufficiency of this statement is not very difficult. Proposition 4.9. Solutions to the Riccati equation

0 2 y = A0(z) + A1(z)y + A2(z)y ,Ai(z) ∈ C(z) can only be branched at a ﬁnite number of points z0.

Proof. We may omit a ﬁnite number of points z0, so let z0 not be a pole of P∞ k/m the Ai(z). Suppose we have a branched solution y = k=A ak(z − z0) ∈ 1/m C{(z − z0) }, m > 1 minimal. Without loss of generality we may suppose n that 0 < A < m, because otherwise we can write y(z) = (z − z0) yb(z) and obtain a similar equation for yb(z) with 0 < A < m. Then the l.h.s. has a pole at z0, while the r.h.s. has not. This is a contradiction, and we conclude that there is only a ﬁnite number of points z0 at which the Riccati equation can have a branched solution.

To prove the necessity, we ﬁrst we need the following lemma.

Lemma 4.10. Let m > 1 and f ∈ C{z, t} with f(0, 0) 6= 0 be given. Then the equation

(ym)0 = f(z, y) (4.1)

1/m 1/m has a solution in C{z } of the form y = cz + ··· , c 6= 0.

38 P∞ i/m 1/m Proof. Suppose we have a solution y = i=1 ciz ∈ C[[z ]] of the differential equation. Then the l.h.s. of equation 4.1 equals

∞   X k X (ym)0 = c ··· c z−1+k/m, m i1 im  k=m i1+···+im=k

m and in particular it starts with the term c1 . On the other hand, the r.h.s. of P∞ i/m equation 4.1 is f(z, i=1 ciz ) and it starts with the term f(0, 0). There- m fore c1 = f(0, 0) 6= 0. Comparing the higher coefficients in equation 4.1 we m−1 find that c1 ca−m+1 equals a polynomial formula in c1, . . . , ca−m, because ca−m+1 has the highest appearing index in the coefficient of a certain a. 1/m Now let y ∈ C[[z ]] be of the form just deduced. One can verify (by brute force) that this series is in fact convergent. More explicitly, we may suppose f(0, 0) = 1 and c = 1. Write (for the moment) z = tm and y = t(1 + h) with h ∈ tC[[t]]. One obtains a differential equation for h of the form d X X t h + mh = c tahb + c hb , dt a,b b a≥1,b≥0 b≥2

a+b b where the coeﬃcients satisfy |ca,b| ≤ R and |cb| ≤ R for some R > 0. After multiplying h and t by suitable constants, we may suppose that R is P n suﬃciently small. Writing h = n≥1 hnt , the recurrence relation for the hn takes the form (n + m)hn = a polynomial in h1, . . . , hn−1 of which the absolute value can be estimated. By induction one shows that |hn| ≤ 1 for all n, and thus h is convergent.

Using this lemma, we can proof the following proposition that almost 0 states that a diﬀerential equation y = R(z, y) ∈ C(z, y) with the APP is a Riccati equation.

Proposition 4.11. Consider a differential field (C(z)(t), d) with t transcendental over C(z) and derivation d given by d(z) = 1, d(t) = A/B, with A, B ∈ C(z)[t] relatively prime and B monic. If B/∈ C(z), then the differential field does not have the APP.

Proof. Assume B/∈ C(z). By Lemma 4.8, we are allowed to make a finite extension C ⊃ C(z) without loss of generality. After adjoining all roots of the polynomial B to the coefficient field C(z), we can suppose B has the m ms form (t−α1) 1 ··· (t−αs) , with αi distinct elements in the newly obtained coefficient field C. After shifting t − α1 to t and adjoining all roots of the polynomial A to C, we may suppose that

s m−1 Y ni t d(t) = β0 (t − βi) , m ≥ 2, 0 6= βi ∈ C, 0 6= ni ∈ Z. i=1

39 Since the extension C ⊃ C(z) is algebraic, there are only finitely many places ramified in the cover C → C(z). Therefore, we can restrict to places of C(z) that are unramified in C. Consider such a place (call it z0), and + choose z0 in C above z0. By Corollary 2.15, this yields an embedding C,→ C({z − z0}). + For every i, there are only finitely many z0 ∈ C for which βi(z0 ) = 0 or + βi(z0 ) = ∞. Omitting these z0, we can assume that all βi in this field have Qs ni order 0. The rational expression β0 i=1(t − βi) can then be written as f(z − z0, t) ∈ C{z − z0, t} with f(0, 0) 6= 0. By Lemma 4.10, the differential m−1 0 1/m equation y y = f(z − z0, y) has a branched solution y = c(z − z0) + 1/m · · · ∈ C{(z − z0) }, c 6= 0. Since we only omitted finitely many values for z0, we find infinitely many points z0 at which the equation has a branched solution. The differential field therefore does not have the APP. This is what we had to prove.

Having proved this proposition, most of the work is done. We have the following corollary that states that the explicit differential equation y0 = R(z, y) ∈ C(z, y) has the APP if and only if it is a Riccati equation. Corollary 4.12. The differential field (C(z)(t), d) with t transcendental over C(z) and C-linear derivation d given by d(z) = 1, d(t) = A/B, with relatively prime A, B ∈ C(z)[t] and B monic in t, has the APP if and only if B = 1 and degtA ≤ 2.

Proof. (⇐=) Assume B = 1 and degtA ≤ 2. Then d(t) = A0(z) + A1(z)z + 2 A2(z)z , with Ai(z) ∈ C(z), is a Riccati equation, which, according to Proposition 4.9, has the APP. (=⇒) Suppose that the diﬀerential ﬁeld has the APP. Then Proposition m 4.11 implies that d(t) = a0 + a1t + ··· + amt ∈ C(z)[t]. On the other hand, −1 2 −1 −m for s = t we have that d(s) = −s (a0 + a1s + ··· + ams ) ∈ C(z)[s], since we can apply Proposition 4.11 with t = s−1. It follows that B = 1 and degtA = m ≤ 2.

4.4 The Generalized Weierstrass Equation

In the last section of Chapter 1 we stated that the first order ordinary differential equations can be classified into two types of equations: The Riccati equation and the generalized Weierstrass equation. In the previous section we proved that the Riccati equation has the APP. In this section we shall prove that the generalized Weierstrass equation also has the APP. We start by proving that the (ordinary) Weierstrass equation admits no solution with branch points and therefore has the APP. Lemma 4.13. The Weierstrass equation

0 2 3 3 2 (y ) = 4y − g2y − g3, g2, g3 ∈ C, ∆ = g2 − 27g3 6= 0,

40 has no solution with a branch point.

Proof. It is well known that the nonconstant solutions of the Weierstrass equation are of the form ℘(z − c), where ℘ is the Weierstrass function and c is an arbitrary constant. Since the Weierstrass function is meromorphic, this proves the lemma (for both claims, see [12, Chapter XIV.2]). To prove this directly, one could use that integrating gives

Z y dy ≡ z + Λ, p 3 a 4y − g2y − g3

3 for a not a zero of 4y − g2y − g3 and a lattice Λ generated by the integrals 3 around two of the zeroes of 4y − g2y − g3. A solution must therefore be a doubly periodic meromorphic function, and these do not have branch points.

The following theorem appeared in a paper of Malmquist [14]. In this paper he gives an analytic proof of the statement, but it is not very readable. The given diﬀerential algebraic proof is much easier.

Theorem 4.14. Let C ⊃ C(z) be a ﬁnite extension. The diﬀerential equation

0 2 3 3 2 ∗ (y ) = a(4y − g2y − g3), g2, g3 ∈ C, ∆ = g2 − 27g3 6= 0, a ∈ C , has the APP.

Proof. Because of Lemma 4.8, we are allowed to make a finite extension of √ our coefficient field C without loss of generality. By adjoining b := 1/ a to d 2 3 C if necessary, we can rewrite the equation as (b dz y) = 4y − g2y − g3. On this new C, define the derivation d by d(f) = bf 0. The equation d(T ) = 1 is a nonhomogeneous linear equation, and therefore it has a solution in some Picard-Vessiot extension L = C(T ) ⊃ C (see [23, Chapter 1.3, proposition 0 1.18] with the equation y1 = 0 1y1). Let T be such a solution. y2 0 0 y2 Assume T is algebraic over C. Then we can adjoin T to our coefficient d 2 3 field and consider the differential equation as an equation ( dT X) = 4X − g2X − g3 over C(T ). This is exactly the situation from Lemma 4.13, and there we saw that there are no branched solutions. Assume T is transcendental over C. Then we have again an equation d 2 3 ( dT X) = 4X − g2X − g3 over the field C(T ). The solutions are ℘(T − c), where ℘ denotes the Weierstrass function. Since the extension C ⊃ C(z) is algebraic, there are only finitely many points ramified in the morphism Z → P. Therefore we can restrict to points z0 ∈ C that are unramified in C. By Corollary 2.15, such a point yields an inclusion C,→ C({z − z0}) of differential fields. There are only finitely many points in C for which the image of b in this field has a pole, so we can restrict to the case that

41 the image of b has no pole. Then the equation d(T ) = bT 0 = 1 has a solution u ∈ C({z − z0}). From this one finds that the solutions ℘(u − c) are compositions of (convergent) Laurent series, and therefore they lie in the field C({z − z0}). This proves that there are at most finitely many points at which the equation can have a branch point and therefore that it has the APP.

Remark 4.15. We shall refer to the equation in Lemma 4.14 as the generalized Weierstrass equation, to distinguish it from the ordinary Weierstrass equation with constant coeﬃcients. In the literature this distinction is often not made. Most of the time when order one equations with the PP are mentioned, one encounters the statement that they classify into the Riccati equation and the Weierstrass equation, without the mentioning of its speciﬁc form.

4.5 Classiﬁcation of the First Order Autonomous Equations

In the previous two sections we proved that the equations of the two classes of our classification have the APP. In this section we shall derive for autonomous equations a geometric criterion equivalent to the APP, and use this criterion to classify the autonomous equations into the two classes of our classification. We include this special case here, because it already contains several ideas from the general case, and because it is much easier. Remark 4.16. That the autonomous case is much easier follows from the following observation. Suppose that y : U → C satisfies the equation F (y0, y) = 0 on some domain in the Riemann sphere. Then, since the equation is autonomous, also ya(z) = y(z + a) is a solution on some domain of the Riemann sphere (for sufficiently small a). Moreover, if y(z) has a branch point at z0, then ya(z) has a branch point at z0 − a. Therefore the existence of a solution with a branch point at z0 immediately implies the existence of a neighborhood of z0 wherein each point is a branch point of some solution. In the autonomous case therefore “F (y0, y) = 0 has a branched solution” is equivalent to “F (y0, y) = 0 has the APP” is equivalent to “F (y0, y) = 0 has the PP”. The following theorem gives a simple criterion for the PP in terms of the differential function field.

0 Theorem 4.17. The autonomous equation F (y , y) = 0,F ∈ K := C(S, T ), has the PP if and only if the derivation d on the associated function ﬁeld C(s, t) has no poles. Proof. The proof is due to Marius van der Put.

42 ‘⇐=’: Suppose that the equation does not have the PP. Then there exists 1/m d a branched solution. Let φ :(K, d) → (C({z }), dz ) be the corresponding homomorphism of diﬀerential ﬁelds for minimal m > 1. φ induces a discrete valuation of the form ∗ v : K −→ Z, f 7−→ m · ord φ(f). By lemma 2.13, this discrete valuation corresponds to a place p whose local ring Op equals {f ∈ K | v(f) ≥ 0}. This means exactly that restricting our 1/m morphism φ to Op gives us a morphism φ|Op : Op → C{z }. Now let π be a local parameter. Then φ(π) is of the form

X n/m φ(π) = bnz , b1 6= 0. n≥1 d Then φ(d(π)) = dz φ(π) ∈/ φ(Op) implying that d(Op) 6⊂ Op. This means that the derivation d has a pole at p. ‘=⇒’: On the other hand, suppose that the derivation d has a pole at the place p. This means that d(Op) 6⊂ Op. By Lemma 2.14, we can identify Op with a subring of C{u}, for u a local parameter at p. Furthermore we can extend d in a unique way to a derivation d : C({u}) → C({u}). Then it d must be of the form d = a(u) du , where a(u) = d(u), since each derivation on C({u}) is determined by its value on u. Furthermore, a(u) has a pole of order 1 − m for a certain m ≥ 2 (it will become clear later why we write down the order in this way). We wish to construct a C-linear homomorphism of differential fields φ : 1/m C({u}) → C({z }). When we have obtained such a homomorphism, we can restrict to Op and then extend to the function field C(s, t) = Frac Op, thus obtaining a branched solution. To get such a homomorphism we would like to send a local parameter 2 1/m π = c1u + c2u + ··· , c1 6= 0, to z . In order for φ to commute with the derivation, we must have d d d 1 1 φ a(u) π = φ(π) = z1/m = z(1−m)/m = φ π1−m , du dz dz m m d 1 1−m so if we can find a local parameter π such that a(u) du π = m π , then we can find the homomorphism φ of differential fields we are looking for as well. d m −1 −1 One rewrites the equation as du (π ) = a(u) . Since a(u) is a convergent power series of order m − 1, there exists a b(u) ∈ C{u} of order m such d −1 that du b(u) = a(u) . Then z := b(u) has an mth root π ∈ C{u} of order one. This is a local parameter with the required property, and we obtain a 1/m homomorphism φ : C({u}) → C({z }) of differential fields. Restricting φ to Ox and extending the result to its field of fractions, we obtain a homomor- 1/m d phism φ :(K, d) → (C({z }), dz ) of differential fields and which yields a branched solution of the differential equation. The equation therefore does not have the PP.

43 0 Corollary 4.18. Let F (y , y) = 0,F ∈ C(S, T ), be an autonomous differential equation with the PP. Then its associated differential function field is either

2 d • (C(z), (a0 + a1z + a2z ) dz ), with a0, a1, a2 ∈ C not all zero, or 2 3 • (C(x, y), d), with y = x + ax + b a (nonsingular) elliptic curve and d ∗ d = Cy dx with C ∈ C . Proof. Assume that the differential function field (K, d) associated to the differential equation F (y0, y) = 0 has the PP. Let X be its corresponding nonsingular projective curve. The proof that the genus g(X) of X is either 0 or 1 is essentially an application of the Riemann-Roch theorem. By Theorem 4.17, there exists a nonzero global section d : O(X) → O(X) of the sheaf DX of derivations, that is, a regular derivation. The sheaf of derivations DX is the dual sheaf of the sheaf of differential 1-forms 1 ΩX/k, which is the canonical sheaf ωX since dim(X) = 1. We therefore know, since D and ωX are linebundles, that D ⊗ ωX is trivial. Hence the deg(D) = −deg(ωX ) = −(2g − 2). However, we must have deg(D) ≤ 0 since it has a global section. Hence the genus of X is either 0 or 1. If g = 0, then X ' P and its function field K ' C(z). In the same way as in Corollary 4.12 we can find that this implies that the derivation is of 2 d d the form d = (a0 + a1z + a2z ) dz . That is, let d = h(z) dz , with h(z) = d h(z) dz z ∈ K be the derivation on K. Since P is a union of its affine open d sets U1 := P\{∞}, U2 = P\{0}, d restricts to derivations f dz : O(U1) → d O(U1) and h dz : O(U2) → O(U2). Since O(U1) = C[z], f ∈ C[z]. Since −1 d −1 −2 −1 2 −1 O(U2) = C[z ], we have h dz −z = hz ∈ C[z ] and h ∈ z C[z ]. 2 d It follows that d = (a0 + a1z + a2z ) dz . If g = 1, then X is an elliptic curve (see [20, Chapter III.6.6, Corollary 4]). Let C(x, y) be its function field, and consider an arbitrary derivation d dx f dx : C(x, y) → C(x, y), f ∈ C(x, y). The differential form y is holomorphic and nowhere vanishing on an elliptic curve X (see [22, Chapter II.4, 1 Example 4.6]). Since X is a curve, ΩX/k is a sheaf of OX -modules of rank 1, and we have an isomorphism of sheaves given by

1 dx OX (U)−→ ˜ Ω (U), f 7−→˜ f ,U ⊂ X open. X/k y On an irreducible projective variety, the only regular functions are the constant functions (see [20, Chapter I.5.2, Theorem 3, Corollary 1]), so we find 1 dx 1 that ΩX/k[X] = C y . On the other hand, the sheaf ΩX/k is the dual sheaf d of the sheaf of derivations DX . Therefore we find that DX [X] = Cy dx . Example 4.19. Consider the curve X defined by F = Sp −T q = 0, (p, q) = 1, q p pq > 1. The rational map P 99K X defined by x 7→ (x , x ) has an inverse

44 b a X 99K P defined by (s, t) 7→ s t , where a and b satisfy ap+bq = 1. Therefore X is birational to P, and consequently its genus is zero. Equip the function field C(s, t) of X with the derivation given by d(t) = s. We claim that X has the PP if and only if |q − p| = 1. Assume that X has the PP. Since g = 0, we know from the previous 2 d corollary that d is of the form (a0 +a1x+a2x ) dx . Using d(t) = s, it follows that d xq = d(t) = a + a x + a x2 xp = a pxp−1 + a pxp + a pxp+1, 0 1 2 dx 0 1 2 which is only possible if q − p ∈ {−1, 0, 1}. Since (p, q) = 1 and pq > 1, it follows that |q − p| = 1. Assume on the other hand that |q − p| = 1. The derivation defined by d(t) = s satisfies xq = d(t) = pxp−1d(x) and is therefore of the form 1 1±1 d d = p x dx . We have seen that this derivation has no poles, and therefore X has the PP. We conclude that X has the PP if and only if |q − p| = 1, confirming the result we found before. Example 4.20. A solution that is meromorphic on the complex plane cor- d responds to a morphism φ :(K, d) → (M(C), dz ), where M(C) denotes d the field of meromorphic functions on C, and dz is the ordinary complex differentiation. Let us determine which solutions correspond to the curves X with the PP. Assume the genus of our curve is zero. Then X = P, K = C(x) is the field 2 d of rational functions, and our derivation d is of the form (a0 +a1x+a2x ) dx . The morphism φ sends x to a certain function f meromorphic on the complex plane. In order for φ to commute with the derivations, we have that d d f = φ a + a x + a x2 x = a + a f + a f 2. dz 0 1 2 dx 0 1 2 It follows that in this case the morphisms φ correspond to meromorphic solutions of an autonomous Riccati equation. Since we saw in Section 4.3 that the Riccati equation has the APP, and since its solutions can only have branch points at the poles of the coefficients, we find that the morphisms φ correspond to all solutions of an autonomous Riccati equation. Now assume that the genus of our curve is one. Then X is an elliptic 2 curve, and its function field C(x, y) is the quotient field of C[X,Y ]/(Y − 3 3 2 4X +g2X+g3) for certain g2, g3 ∈ C satisfying g2 −27g3 6= 0. The derivation d on C(x, y) is of the form Cy dx with 0 6= C ∈ C, so it sends x to Cy. The morphism φ sends x to a certain function f meromorphic on the complex plane. In order for φ to commute with the derivation, we have that

d 2 d 2 f = φ(Cy x) = C2(4f 3 − g f − g ),C 6= 0 6= g3 − 27g2. dz dx 2 3 2 3

45 It follows that the morphisms φ correspond to the meromorphic solutions of the Weierstrass equation. As we have seen in Section 4.4, the nonconstant solutions of this equation are translated Weierstrass ℘-functions, and these are all meromorphic. Therefore the morphisms φ correspond to the solutions of a Weierstrass equation. We conclude that the only global solutions of autonomous ﬁrst order equations with the PP are solutions of an autonomous Riccati equation and translates of a Weierstrass ℘-function.

4.6 Classiﬁcation of the First Order Equations

This section uses a lot more notions from algebraic geometry than was de- scribed in Chapter 2 and which has been used so far. For all new notions in this chapter we refer to the book Algebraic Geometry by Hartshorne [9]. In this section we show that the results from the previous section can be generalized to the general case. We start by noting that we can consider branched solutions as homomorphisms from (a subring of) the function field to some field of Puiseux series. Next we use this to prove that a differential field with the APP must have a regular derivation. In a subsequent proposition we assume that a differential field has a regular derivation and find that it can only be trivial or correspond to a Riccati or generalized Weierstrass equation. As we have seen in Sections 4.3 and 4.4, these equations have the APP. This then concludes our classification. Let Z/C be the nonsingular projective curve (considered as a Riemann surface) associated to the field extension C ⊃ C, let πp be a local parameter at a place p of C , and let C{πp} denote the analytic local ring at p. The 1/m question is whether there exists a solution y ∈ C({πp }) that does not belong to C({πp}). To answer this question, we shall rephrase it in terms of differential homomorphisms into the field of Puiseux series. We explicitly construct such morphisms, starting from the inclusion OZ,p ,→ C{πp} of differential rings we have by Lemma 2.14. It turns out that there are two cases to distinguish for a solution y ∈ C({z − z0}): it can be algebraic over C(z) or transcendent. In the case that y is transcendent, it corresponds to 1/m d a homomorphism of differential fields (K, d) → (C({(z − z0) }), dz ), with m > 1 minimal. In the case that y is algebraic over C(z), it corresponds 1/m d to a homomorphism of differential rings (OX,q, d) → (C({(z − z0) }), dz ), with m > 1 minimal for some maximal differential subring OX,q ⊂ K. The following lemma is the first step in this process.

1/m Lemma 4.21. Let p be a place of C, and let y ∈ C({πp }), m > 1 minimal, be a solution of F (y0, y) = 0 satisfying ∆(y0, y) 6= 0, where ∆ := dF ds (s, t). Then there exists a unique homomorphism of diﬀerential rings 1/m ψy : C[s, t]∆ → C({πp }) that restricts to the power series expansion

46 OZ,p ,→ C{πp} at the local ring of p, where C[s, t]∆ denotes the localization at ∆.

Proof. In what follows, write π for πp. By Lemma 2.24, the inclusion of d differential rings (OZ,p, d) ,→ (C{π}, dπ ) induces an inclusion ι :(C, d) ,→ d (C({π}), dπ ) of differential fields. On the extension C[s, t] := C[S, T ]/(F ) ⊃ C of rings define a C-linear mapping d : C[s, t] → C(s, t) by d(z) = 1, d(t) = s and d(s) by

n n X k X k−1 0 = d(F (s, t)) = d(ak(t))s + kak(t)s d(s), k=0 k=1 which is well-defined since F is irreducible. Then d coincides with our derivation d on C, but C[s, t] is in general not closed under d, only because the reciprocal of ∆ is in general not an element of C[s, t]. The localization −1 C[s, t]∆ is equal to the ring C[s, t, ∆ ], which is closed under d. Therefore C[s, t]∆ is a differential ring. 1/m Then ι extends to a mapping ψy : C[s, t] → C({π }) given by ψy(t) = d y, ψy(s) = dz y. Since ψy(∆) 6= 0, we may extend our ψy to a homomorphism 1/m −1 −1 C[s, t]∆ → C({π }) of differential rings, by defining ψy(∆ ) = ψy(∆) . This gives us the desired homomorphism of differential rings.

Remark 4.22. The lemma does not consider the case where ∆(y0, y) = 0. Since there are only finitely many such y ∈ C({π}) that are a solution of the differential equation as well, these can be neglected if we are only interested in the question of whether there are infinitely many branch points.

1/e Lemma 4.23. ψy : C[s, t]∆ → C({π }) has nontrivial kernel if and only if y is algebraic over C.

Proof. (⇐=): First suppose y is algebraic over C. Then there exists a 0 6= F ∈ C[Y ] such that F (y) = 0, and 0 6= F (t) ∈ ker ψy. This means that ψy has a nontrivial kernel. (=⇒): On the other hand, suppose that ker ψy 6= (0). Since ψy maps into a domain, this is a prime ideal. Because x is a nonsingular point, the local ring C[s, t]∆ is regular [9, Chapter 1, Theorem 5.1] and consequently has dimension 1. We therefore have that ker ψy is the only nonzero prime ideal, implying that ker ψy is the unique maximal ideal of C[s, t]∆. It follows that 1/e ψy induces a morphism ψy : C[s, t]∆/m → C({π }) of differential fields. By the Weak Nullstellensatz C[s, t]∆/m ⊃ C is a finite field extension, and it follows that y is algebraic over C.

Remark 4.24. Suppose ker ψy 6= (0). As we saw in the above proof, ker ψy is a maximal ideal. Moreover, by construction ψy commutes with taking derivatives, and therefore this maximal ideal is closed under d.

47 Remark 4.25. Because Z is chosen to be nonsingular, we have a morphism Z → P that satisﬁes the conditions of Lemma 2.16. A local parameter π on 1/m the covering Z therefore corresponds to a function (z − z0) on an open 1/m neighborhood of z0 in P. In what follows, we shall identify π and (z−z0) . The following lemma shows that in case y is transcendental we can go one step further and extend ψ to the whole function ﬁeld K.

Lemma 4.26. Let ψy be the above homomorphism. If y is transcendental over C, then ψy extends uniquely to a homomorphism

1/e ψy : K −→ C({(z − z1) }) of differential fields, where K is the field of fractions of C[s, t], and e is a multiple of m.

Proof. Suppose y is transcendental over C. By Lemma 4.23, y transcendental implies that ker ψy = (0). Therefore, using Proposition 2.19, the derivation d : C[s, t]∆ → C[s, t]∆ can be extended in a unique way to a derivation on K, thus obtaining a differential field (K, d). By Lemma 2.24, the inclu- 1/e sion ψy : C[s, t]∆ ,→ C({(z − z1) }) of differential rings can uniquely be 1/e d extended to an inclusion ψy :(K, d) ,→ (C({(z − z1) }), dz ) of differential fields, with e > 1 minimal. This is what we wanted to prove.

Before we can proceed to state our main results, we need one more lemma that essentially states that the regularity of the derivation is invariant under extensions of our ground ﬁeld K.

Lemma 4.27. Let X be a nonsingular curve over K with a derivation D on its function ﬁeld. The statement ‘the local ring OX,p is not closed under D’ is equivalent to the statement ‘the local ring OL⊗K X,p’ is not closed under D’.

Proof. We have the following commutative diagram

...... OX,p .... ObX,p K[[π]] ...... L[[π]] OL⊗K X,p ObL⊗K X,p where ObX,p = K[[π]] and ObL⊗K X,p = L[[π]] denote the completions of the local rings. Because of this, we can identify the local rings with their power series. We see that OX,p is not closed under the diﬀerentiation if and only if D(π) is a Laurent series starting at a term with negative exponent, and the same holds for OL⊗K X,p = L[[π]]. The statement of the theorem follows directly from this.

48 The following proposition shows that also in the general case, the APP implies our condition on the differential function field found in the previous section. The structure of the proof is as follows. We start with a place p of K at which the derivation has a pole. After a finite extension of C, we can assume it to be rational. K is the function field of a normal projective surface X /C, and we obtain a rational map X 99K Z together with a rational section Z 99K X . Viewing X /C and Z/C both as complex manifolds, we can locally explicitly write down these rational maps between U ⊂ C and 2 V ⊂ C , and find explicitly a local parameter u. The hypothesis then m implies that d(u) ∈/ C[[u]], and this induces a differential equation d(u ) = g(z − z1, u) ∈ C{z − z1, u} with g(0, 0) 6= 0, which, as we saw in Lemma

4.10, has a branched solution hz1 . It then follows that in all but a finite number of cases, hz1 induces a solution to our original differential equation branched at the point z1. Proposition 4.28. Suppose that the derivation d is not regular. Then the differential equation does not have the APP. Proof. Let p be a place of K over C at which the derivation has a pole. By Lemma 4.8 and Lemma 4.27, both the hypothesis and the statement we want to prove are stable under finite extensions of C. We may therefore suppose that the closed point p is rational over C. The field K can also be seen as the function field of some normal projective surface X over C. The inclusion C,→ K induces a rational map X 99K Z, where Z is the absolutely irreducible, nonsingular and projective curve over C with function field C. The assumption that p is a C-rational point of X induces a ‘rational section’ of the above rational map. In the analytic category, avoiding singularities etc., one can generically describe this rational map and section by the following. Let U ⊂ Z be a disk identified with {z ∈ C : |z − z0| < ε}, and let V ⊂ X be a multidisk identified with {z ∈ C : |z − z0| < ε} × {u ∈ C : |u| < ε}. The rational map is just (z, u) 7→ z, and the rational section is z 7→ (z, 0). Furthermore, u = 0 is the restriction to V of the divisor [p] on X . We note that u is a local parameter at p, and the completion ObX,p equals C[[u]], since p is nonsingular. By Lemma 2.14, we have an inclusion OX,p ,→ C[[u]] of rings. It is given that OX,p is not invariant under d, and therefore d(u) ∈/ C[[u]]. Hence

−m+1 d(u) = a−m+1u + ··· , 0 6= a−m+1 ∈ C, m ≥ 2, and consequently d(um) ∈ C[[u]] with nonzero constant term. In analytic terms, d(um) is a nonzero holomorphic function on V . By choosing a smaller ε if necessary, we can shrink our U and V , and we may suppose that d(um) has no zeros on V . m Now take an arbitrary point z1 ∈ U. We have d(u ) = g(z − z1, u) ∈ C{z − z1, u} with g(0, 0) 6= 0. According to Lemma 4.10, the diﬀerential

49 X [p]

Z [p]

U C V

Figure 4.1: A schematic representation of the situation in the proof of Propo- sition 4.28.

m 1/m equation d(h ) = g(z −z1, h) has a solution of the form h = h1(z −z1) + 1/m · · · ∈ C{(z − z1) } with h1 6= 0. Using this solution, we get a homomor- 1/m phism ψ : C{z − z1, u} → C{(z − z1) }, given by z − z1 7→ z − z1, u 7→ h, which in turn induces a homomorphism of differential rings ψ : C[u, d(u)] → 1/m C({(z − z1) }). Suppose h is transcendental over C. Then Lemma 4.26 implies that ψ can be extended to a homomorphism of differential fields d ψ :(K, d) −→ ( {(z − z )1/e}, ), C 1 dz where e is some multiple of m. That is, we have found a branched transcendental solution. Suppose h is algebraic over C. Then ψ is a homomorphism

1/e d ψ :(OX,q, d) −→ ( {(z − z1) }, ) C dz of diﬀerential rings, where OX,q is a local and maximal diﬀerential subring of K, and e is a multiple of m. We distinguish two cases. Suppose that t ∈ OX,q. Then d(t) ∈ OX,q, therefore C[s, t] ⊂ OX,q. Consequently the restriction of ψ to the subring C[s, t] gives a branched algebraic solution. Finally suppose that t∈ / OX,q. Then, since OX,q is a valuation ring, −1 t lies in the maximal ideal mX,q of OX,q, because this ideal consists of the noninvertible elements of OX,q. By Remark 4.24, ker ψ is the maximal

50 ideal mX,q of OX,q and is closed under the derivation. The ﬁrst isomorphism theorem of rings then implies that ψ has a unique factorization

π ... OX,q ...... OX,q/mX,q ...... f ...... ψ ...... 1/e C({(z − z1) }) where π is the canonical projection x 7→ x + mX,q, and f is a differential field inclusion. The place of C corresponding to z1 is ramified in OX,q/mX,q. Using the Weak Nullstellensatz, we find that the extension OX,q/mX,q ⊃ C is a finite extension, and therefore there are only finitely many of these places. On the other hand, there are only finitely many q with t∈ / OX,q. Therefore the number of these points z1 is finite. Thus, with the exception of the finite number of points just encountered, we find that the differential equation has a branched solution at z1. We have once more shown that the differential equation does not have the APP.

The other direction, that a differential field with a regular derivation has the APP, is a direct consequence of the next proposition. The proof actually states something stronger. When the derivation is regular, we can make a finite extension of our coefficient field and end up with a curve that also comes from some curve over the complex numbers to which some base extension has been applied. In particular this holds for the differential fields corresponding to the Riccati equation and the generalized Weierstrass equation, and we find that these differential fields are the only ones with a nontrivial regular derivation (recall that a trivial derivation would mean s = d(t) = 0, a case we already considered in Section 4.1). On the other hand, we saw in the Sections 4.3 and 4.4 that these have the APP. Therefore it follows that differential fields with a regular derivation have the APP. The proof of the general statement is complicated, because it uses a lot of algebraic geometry. However, if we assume that our differential field comes from a curve that is either hyperelliptic or has genus zero or one, then the proof does not transcend the prerequisites needed so far. Proposition 4.29. Suppose that the derivation d on K is regular. Then there exists a finite extension L ⊃ C and a nonsingular curve X0/C such that X ×C L ' X0 ×C L. Moreover, the only differential function fields with a nontrivial derivation are the differential function fields associated to the Riccati and the generalized Weierstrass equation. Proof. Clearly the statement, and by Lemma 4.27 also the hypothesis, is stable under a finite extension of the base field C. We distinguish four cases:

51 Case 1: g = 0. If the genus of X is zero, then its function field K is C(t). After applying any base extension L ⊃ C (or taking L = C), we get a curve X ×C L with function field L(t). On the other hand, if we start with the curve X0 := P over C and apply a base extension L ⊃ C, we obtain a curve with function field L(t) as well. We examined the case g = 0 at the beginning of the chapter and found that the derivation d should be given by 2 d(t) = a0 + a1t + a2t , with ai ∈ C not all zero. Case 2: g = 1. Suppose that the genus of X is one. After taking a finite extension of C, we may suppose that X is in Legendre form, that is, given by an affine equation

X : s2 = t(t − 1)(t − a), 0, 1 6= a ∈ C.

The function field of X is then of the form K = C(s, t), where s and t are related by the above equation. After applying a base extension L ⊃ C, we obtain a curve X ×C L with function field L(s, t). Using the assumption on d for t and for all points, except the point at infinity, we find that d(t) = A(t) + B(t)s with A(t),B(t) ∈ C[t]. To obtain an additional condition on d, we use the remaining information from the assumption: the local ring OX,∞ is closed under d. To make calculations in this ring, we write OX,∞ in its power series expansion. Let π be a local parameter at ∞. The completion of OX,∞ has the form

−1 2 −3p 2 2 ObX,∞ = C[[π]], t = π , s = π (1 − π )(1 − aπ ). Since 2d(π) = −π3 A(π−2) + B(π−2)π−3p(1 − π2)(1 − aπ2) has to be an element of C[[π]], it follows that A(t) = a0 + a1t and that B(t) = b0. We did not yet use that d(s) ∈ C[s, t]. The expression 2sd(s) = d(t(t − 1)(t − a)) equals

1 1 1 d(a) t(t − 1)(t − a) + + (a + a t + b s) − , t t − 1 t − a 0 1 0 t − a and therefore it follows from the condition d(s) ∈ C[s, t] that

1 1 1 d(a) t(t − 1)(t − a) + + (a + a t) − t t − 1 t − a 0 1 t − a is an element of sC[s, t]∩C[t] = t(t−1)(t−a)C[t]. Hence a0 = a1 = d(a) = 0. Therefore a ∈ C, and we can take X0/C to be the curve

2 X0 : s = t(t − 1)(t − a), 0, 1 6= a ∈ C and L = C. This proves the statement in the case that g = 1, and we ﬁnd ∗ moreover that d has the special form d(t) = b0s, for some b0 ∈ C .

52 Case 3: X is a hyperelliptic curve. Suppose that X is a hyperelliptic curve. The proof is essentially the same as in the proof of the elliptic case. After taking a ﬁnite extension of C, we may suppose that an aﬃne equation of X has the form

2 X : s = Q = t(t − 1)(t − c1) ··· (t − cm), m > 1 odd, and 0, 1, ci ∈ C distinct elements. The function field of X is then of the form K = C(s, t), where s and t are related by the above equation. After applying a base extension L ⊃ C, we obtain a curve X ×C L with function field L(s, t). Again because of the assumption on d for t and for all points, except the point at infinity, d(t) = A(t)+B(t)s with A(t),B(t) ∈ K[t]. The completion of OX,∞ can be written as

m −1 2 −m−2p 2 Y p 2 C[[π]], t = π , s = π 1 − π 1 − ciπ . i=1 In the same way as before, we have that

m ! 3 −2 −2 −m−2p 2 Y p 2 −2d(π) = π A(π ) + B(π )π 1 − π 1 − ciπ , i=1 from which it follows that A(t) = a0+a1t and B(t) = 0, if d(π) ought to lie in C[[π]]. As before, the condition d(s) ∈ C[s, t] implies that 2sd(s) = d(Q) ∈ sC[s, t] ∩ C[t] = QC[t]. Now d(Q) ∈ QC[t] implies that a0 = a1 = d(ci) = 0. Therefore ci ∈ C, and we can take X0/C to be the curve 2 X : s = Q = t(t − 1)(t − c1) ··· (t − cm), m > 1 odd, 0, 1 6= ci ∈ C and L = C. This proves the statement in the hyperelliptic case, and we ﬁnd moreover that d has the special form d(t) = d(s) = 0. That is, d is trivial. Case 4: g ≥ 2, and X is not a hyperelliptic curve. We wish to extend the derivation d : K → K to a C-linear morphism of sheaves X X ∇ :ΩX/C −→ ΩX/C , fidgi 7−→ (d(fi)dgi + fidd(gi)) i i

(be careful not to mix up the derivation d and the diﬀerential dgi). Consider an aﬃne open subset U ⊂ X. Write O(U) = C[X1,...,Xn]/ (f1, . . . , fs) = + C[x1, . . . , xn]. One can lift d to a derivation d : C[X1,...,Xn] → C(X1,..., + Xn) having the property that d (f1, . . . fs) ⊂ (f1, . . . , fs). Let V be the free O(U)-module with basis dx1,... dxn, and let W be the O(U)-submodule

W = span {dfi | i = 1, . . . , s} = span {df | f ∈ (f1, . . . , fs)}.

Then ΩX/C (U) := V/W .

53 Deﬁne X X ∇U : V −→ V, hidxi 7−→ (d(hi)dxi + hidd(xi)) . i i

To transfer this deﬁnition to ΩX/C (U) one has to check that ∇U (W ) ⊂ W . + This follows directly from d (f1, . . . , fs) ⊂ (f1, . . . , fs). There results a ∇U :ΩX/C (U) → ΩX/C (U) of the required form. It is not hard to see that the ∇U commute with the restrictions of the sheaf ΩX/C . That is, for U1 ⊂ U2 the diagram

∇U 2 ... ΩX/C (U2) ...... ΩX/C (U2) ...... U . . U 2 . . 2 ρ . . ρ U . . U 1 . . 1 ...... ΩX/C (U1) ...... ΩX/C (U1) ∇U1 commutes. The ∇U therefore deﬁne a morphism of sheaves ∇. 0 Now the C-vector space H (X, ΩX/C ) is invariant under ∇, making 0 H (X, ΩX/C ) into a diﬀerential module over C. The same holds for the 0 s symmetric powers H (X, Sym ΩX/C ). Thus we have found line bundles L that are invariant under d. That is, d maps H0(U, L ) to itself for any open U ⊂ X for such L . Consider such a line bundle L that is at the same time very ample. Then it is well known that   M 0 s X = Proj  H (X, Sym L ) . s≥0

Each C-vector space H0(X, SymsL ) has an action ∇ on it, which makes it into a differential module. We take a Picard-Vessiot extension U ⊃ C which 0 s trivializes these differential modules. This means that U ⊗C H (X, Sym L ) has a basis {e1, . . . , e1} such that ∇e1 = ··· = ∇en = 0. In fact, it suffices to trivialize the differential module H0(X, L ), since the H0(X, SymsL ) are images of the sth symmetric powers of H0(X, L ). That is, we have a mapping

SymsH0(X, L ) −→ H0(X, SymsL ).

Then X ×C U = Proj U ⊗C H, where H = ⊕s≥0Hs is the homogeneous 0 s C-algebra, generated by H1, with Hs = ker(∇,H (X ×C U, Sym Lext)) for all s ≥ 0. If we put X0 = Proj(H), then X ×C U ' X0 ×C U. It is well known that in this case there exists also a ﬁnite extension L ⊃ C such that

X ×C L ' X0 ×C L.

54 We add here that for g ≥ 2, the derivation d has a special form. Write (after a finite extension of C) K as C(s, t), where C(s, t) is the function field of X0. Then d is zero on C(s, t). This means that the corresponding differential equation is trivial.

Remark 4.30. Since for the Riccati and the generalized Weierstrass equation there do not exist infinitely many points at which there exists a local solution with an essential singularity, we see that the APP is much stronger than we thought at first. A differential equation with the APP therefore also lacks any movable essential singularities. Corollary 4.31. Suppose that the differential equation F (z, y0, y) = 0 de- d fines a differential field extension (K, d) ⊃ (C, dz ) such that d is regular. Then F (z, y0, y) = 0 has the APP. Furthermore, there are no movable singularities other than poles. Proof. For the case that the genus of K/C is 0 or 1, we gave explicit calculations at the beginning of this chapter. For genus bigger than one, it follows from the proof of the previous proposition that we may suppose that K = C(s, t), where C(s, t) is a differential function field satisfying d(s) = d(t) = 0. A branched complex analytic solution y to F (z, y0, y) = 0 on some domain in the complex plane corresponds to a homomorphism 1/e ψy : C(s, t) → C({(z − z0) }) of differential fields. Since d 0 = ψ (d(a)) = ψ (a), a ∈ {s, t}, y dz y we find that ψy sends s and t to constants (that is, to elements of C). The elements of C are sent to algebraic functions with fixed ramification points. The only movable singularities are therefore poles. This proves the corollary.

Combining Proposition 4.28 and Corollary 4.31, we arrive at the main theorem of this thesis.

Theorem 4.32. Let C ⊃ C(z) be a ﬁnite extension. Suppose that the diﬀerential equation

F (y0, y) = 0, with F ∈ C[S, T ] absolutely irreducible and both S and T appearing in F , d defines a differential field extension (K, d) ⊃ (C, dz ) as above. Then the differential equation has no movable singularities if and only if the derivation d is regular. Furthermore the differential fields for which the corresponding differential equation has no movable singularities are classified according to the genus of the associated curve, and are precisely

2 d • Riccati type: (C(x), (a0 + a1x + a2x ) dx ), a0, a1, a2 not all zero.

55 d • Weierstrass type: (C(x, y), cy dx , with C(x, y) the function ﬁeld of an elliptic curve and c 6= 0.

• Algebraic type: (C(x, y), d), with d the trivial derivation, d(t) = d(s) = 0.

56 CHAPTER 5 Discussion

n this ﬁnal chapter we shall discuss the material presented in the previous I chapters. First in Section 5.1 we summarize what was done in this thesis, and second in Section 5.2 we discuss what can be done in the future.

5.1 Conclusions

The following details were given for the classical and a modern theory of first order differential equations with the PainlevéProperty. 1. The classical case: (a) An overview was given of the classical literature. The literature was often found to be incomplete and sometimes incorrect. The authors whose work has been discussed were • for primary sources: Briot and Bouquet, Fuchs, Poincaréand Painlevé; • for secondary sources: Ince, Malmquist, Bieberbach and Hille. (b) The classical criterion for the PainlevéProperty was restated for the autonomous and the general case. In the autonomous case a detailed proof was given. (c) The classical criterion for the PainlevéProperty for autonomous equations was turned into an algorithm and implemented in Maple. 2. The modern case: (a) A geometric criterion that appeared in a book of Matsuda was shown to be equivalent to the (Algebraic) PainlevéProperty. (b) A modern classification was carried out and written down in de- tail.

57 5.2 Future Work

Chapter Modern PainlevéTheory contains a modern classification of the first order ordinary differential equations with the PainlevéProperty. On the other hand Chapter Classical PainlevéTheory does not yet contain a complete overview of the classical literature, and there are many more classical theorems of which a readable version belongs in this chapter. More precisely, the following improvements can be made.

1. Read the work of Painlev´e[16],[17], Poincar´e[18], Forsyth [5] and Golubew [8], and summarize precisely what they have done.

2. Restate and reprove other important classical theorems from Briot and Bouquet, Fuchs and Painlev´e,for instance the classical classiﬁcation theorem.

Apart from this, more details can be given on the material presented in Chapter Modern Painlev´eTheory, and it might generalize to other classes of diﬀerential equations.

1. Examine if the content of Chapter 4 generalizes to higher order or partial diﬀerential equations.

2. The relation between Fuchs’s Criterion and Matsuda’s criterion can be made more explicit. Using the Pl¨ucker formulae one can probably show that the Fuchs’s Criterion implies that the genus of the associated curve is zero.

3. Just as for first order PainlevéTheory, an overview of the second order theory has never been written down in a proper way. This thesis might serve as a prototype for an overview of the second order Painlevé Theory, which is much more important than the first order case.

58 APPENDIX A Implementation of a Painlev´eTest in Maple

This appendix contains an implementation of the algorithm from Section 3.3 in Maple. This implementation uses the puiseux-function from the algcurves- package, which was written by Mark van Hoeij and comes standard with Maple. However, it is not loaded by default. Therefore, before executing the procedure below one must activate this package with the Maple-command with(algcurves);. IsPainleve := proc(F) local i , Finf, Lp, Lc, C, P, z, n, o;

# Initial values. Lp := {}:

# Check if the input satisﬁes the conditions of the test . if not(irreduc(F)) then ERROR(‘The test for irreducibility fails .‘) ; end if ;

# Check if the condition A1 holds. for i from 0 to degree(F, y) do if degree(coeﬀ (F, y, degree(F, y) − i), x) > 2∗i then return false ; end if ; end do;

# Compute the form of the equation at inﬁnity.

59 Finf := subs( x = 1/X, y = −Y / Xˆ2, F); Finf := subs( X = x, Y = y, normal(denom(Finf) ∗ Finf));

# If there are branch points at inﬁnity , then put the # corresponding Puiseux series in the list Lp. if (subs(x = 0, discrim(Finf, y)) = 0) then Lp := puiseux(Finf, x = 0, y, 2, T); end if ;

# Make a set Lc of all critical points of F. Lc := {solve(discrim(F, y) = 0)}; #print(Lc);

# Add the Puiseux series in each branch point to Lp. for C in Lc do Lp := Lp union puiseux(F, x = C, y, 2, T); end do;

# Check the properties A2 and A3 for each Puiseux series. for P in Lp do n := degree(rhs(P[1]) , T); # degree of branching z := subs(T = 0, rhs(P[2])); # constant term o := ldegree(rhs(P[2])) / n; # order

if (z <> 0 and n > 1) then return false ; end if ;

if (z = 0 and o < 1 and o <> (n−1)/n) then return false ; end if ; end do;

return true; end proc:

60 APPENDIX B The Painlev´eProperty in Physics

This appendix contains a list of areas in physics in which equations with the PP appear, mostly second order equations and some partial diﬀerential equations.

• Asymptotics of nonlinear evolution equations

• Correlation functions of the XY model

• Two-dimensional Ising model

• Statistical mechanics

• Random matrix models

• Quantum gravity and quantum ﬁeld theory

• Topological ﬁeld theory (WDVV equations)

• Solutions of the SDYM and stationary axisymmetric Einstein equations

• Surfaces with constant negative curvature

• General relativity

• Plasma physics

• Resonant oscillations in shallow water

• Convective ﬂows with viscous dissipation

• G¨ortler vortices in boundary layers

61 • Nonlinear waves

• Hele-shaw problems

• Polyelectrolytes and electrolysis

• Superconductivity

• Bose-Einstein condensation

• Nonlinear optics and ﬁbre optics

• Stimulated Raman scattering

62 BIBLIOGRAPHY

[1] Bieberbach, L. Theorie der Gew¨ohnlichen Diﬀerentialgleichungen: auf Funktionentheoretische Grundlage Dargestellt. Springer, Berlin, 1953.

[2] Boyer, C., and Merzbach, U. A History of Mathematics, 2nd ed. Wiley, New York, 1991.

[3] Briot, C., and Bouquet, J. Mémoiresur l’intégration des équations différentielles au moyen des fonctions elliptiques. Journal de l’Ecole´ (Imperiale) Polytechnique 36 (1856), 199–254.

[4] Chen, G., and Ma, Y. Algorithmic reduction and rational general solutions of ﬁrst order algebraic diﬀerential equations, 2005.

[5] Forsyth, A. Theory of Diﬀerential Equations: Ordinary Equations, Not Linear, vol. 2. Cambridge University Press, London, 1906.

[6] Fuchs, L. über differentialgleichungen, deren integrale feste verzweigungspunkte besitzen. Königlichen Preussischen Akademie der Wis- senschaften 32 (1884), 699–719.

[7] Fulton, W. Algebraic Curves. W.A. Benjamin, Inc, New York, 1969.

[8] Golubew, W. W. Vorlesungen ber Diﬀerentialgleichungen im Kom- plexen. VEB Deutscher Verlag der Wissenschaften, Berlin, 1958.

[9] Hartshorne, R. Algebraic Geometry. Springer-Verlag, New York, 1977.

[10] Hille, E. Ordinary Diﬀerential Equations in the Complex Domain. Wiley-Interscience, New York, 1976.

63 [11] Ince, E. Ordinary Differential Equations. Dover Publications, Inc, New York, 1956. [12] Lang, S. Complex Analysis. Springer-Verlag, New York, 1999. [13] Malmquist, J. Sur les fonctions àun nombre fini de branches definies par les equations différentielles du premier ordre. Acta Math. 36 (1913), 297–343. [14] Malmquist, J. Sur les fonctions áun nombre fini de branches satis- faisant áune équation différentielle du premier ordre. Acta Math. 74 (1941), 175–196. [15] Matsuda, M. First Order Algebraic Differential Equations: A Dif- ferential Algebraic Approach. Lecture Notes in Mathematics. Springer- Verlag, Berlin, 1980. [16] Painleve,´ P. Sur les equations différentielles du premier ordre. Comptes-Rendus Acad. Sc. Paris 107 (1888), 221–224, 320–323, 724– 726. [17] Painleve,´ P. Le¸cons sur la Théorie Analytique des Equations´ Différentielles, Professées a Stockholm. Hermann, Paris, 1895. Reprinted, Oeuvres de Paul Painlevé, vol. I (Editions du CNRS, Paris, 1973). [18] Poincare,´ H. Sur un théorème de m. fuchs. Acta Mathematica 7 (1885), 1–32. [19] Puiseux, V. Recherches sur les fonctions algébriques. Journal de MathématiquesPures et Appliquées15 (1850), 365–480. [20] Shafarevich, I. Basic Algebraic Geometry 1. Springer-Verlag, Berlin Heidelberg, 1994. [21] Shafarevich, I. Basic Algebraic Geometry 2. Springer-Verlag, Berlin Heidelberg, 1994. [22] Silverman, J. The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986. [23] van der Put, M., and Singer, M. Galois Theory of Linear Differen- tial Equations. Lecture Notes in Mathematics. Springer-Verlag, Berlin Heidelberg, 2003. [24] van der Waerden, B. Einführung in die Algebraische Geometrie. Springer-Verlag, Berlin, 1973. [25] Walker, R. Algebraic Curves. Princeton University Press, New Jersey, 1950.

64 INDEX

A Discrete valuation ...... 43

Absolutely irreducible ...... 35 F Algebraic function...... 10 Analytic function...... 11 Field of constants ...... 17 Associated curve ...... 8 First order equation ...... 6 Associated differential field . . . . . 8 Fixed problematic point ...... 4 Autonomous equation ...... 5 Forsyth ...... 58 Franco-Prussian War ...... 33 B Fuchs ...... 21, 33, 57 Fuchs’s Criterion ...... 7, 22, 32 Bieberbach...... 23 f., 57 Function field Bouquet ...... 20, 26, 57 algebraic ...... 17 Branch point ...... 3, 15 differential ...... 17 Briot ...... 20, 26, 57 G C Gambier ...... 23 Chen ...... 32 Genus ...... 44 Coefficient field...... 35 Golubew ...... 58 Constant ...... 17 Curve H hyperelliptic ...... 53 Hille ...... 23 f., 57 D Holomorphic function ...... 11

Degree of a place ...... 14 I Derivation ...... 8, 16 regular ...... 8 Ince ...... 23, 57 Differential field ...... 16 L Differential field extension . . . . . 17 Differential ring ...... 16 Laurent series

65 convergent ...... 10 Riemann surface ...... 3, 46 formal ...... 10 Riemann-Roch Local parameter...... 43 theorem...... 44 Local ring ...... 14 analytic ...... 46 S regular ...... 47 Saito ...... 23 M Singularity ...... 4 essential ...... 4, 55 Ma...... 32 Solution Malmquist ...... 8, 23, 41, 57 algebraic ...... 46 Matsuda ...... 8, 33 transcendental...... 46, 48 Meromorphic function. .11, 35, 45 Subfield Movable problematic point . . . . . 4 differential ...... 17 Multivaluedness ...... 3, 19 T O Transformation ...... 7 Ostrowski ...... 12 V P Valuation ring...... 13 Painlevé...... 8, 24, 57 Valuative criterion of properness PainlevéProperty ...... 4, 37 14 Algebraic ...... 37 Place...... 14 W Poincaré...... 2, 22, 24, 57 Waerden, van der ...... 12 Pole ...... 4, 14 Weak Nullstellensatz...... 47, 51 Power series Weierstrass ...... 7 convergent ...... 10 Weierstrass ℘-function...... 41 formal ...... 10 Weierstrass equation Problematic point...... 2, 4 generalized ...... 8, 40 Puiseux ...... 19 ordinary ...... 40 Puiseux series convergent...... 11, 35 formal ...... 11 Puiseux Theorem...... 12 R

Radius of convergence ...... 10 Ramiﬁcation ...... 15 Rational point ...... 14, 49 Residue ﬁeld ...... 14 Riccati ...... 7 Riccati equation ...... 7, 38

66 List of Symbols

PP PainlevéProperty ...... 4 P Riemann sphere ...... 4 gcd Greatest common divisor ...... 5 R Commutative ring without zero divisors ...... 9 k, L Fields ...... 9 C Complex numbers ...... 9 k Algebraic closure of a field k ...... 10 N0 Natural numbers and zero ...... 10 Z Integers 0, ±1, ±2,...... 10 R[X] Polynomials with coefficients in R ...... 10 k(X) Rational functions with coefficients in k ...... 10 k(X) Algebraic functions over k(X) ...... 10 k[[X]] Formal power series with coefficients in k ...... 10 K{{X}} Convergent power series with coefficients in k ...... 10 k((X)) Formal Laurent series with coefficients in k ...... 10 k({X}) Convergent Laurent series with coefficients in k ...... 10 N Natural numbers 1, 2, 3,...... 11 lim −→ Direct limit ...... 11 lim k((X1/d)) −→ Formal Puiseux series with coefficients in k ...... 11 lim k({X1/d}) Convergent Puiseux series with coefficients in k ...... 11 −→ M(U) Meromorphic functions on an open set U ⊂ C ...... 11 C,K,F Differential fields ...... 13 ∆ Discriminant of a polynomial equation ...... 13 p Place of a curve ...... 14 O, Op Valuation ring or local ring at p ...... 14 X,Y,Z Algebraic curves ...... 14 Kp Residue field at a place p in a field K ...... 14 deg p Degree of a place p,[Kp : K] ...... 14

67 d Derivation ...... 16 A1 − A4 Conditions from Briot and Bouquet’s Theorem ...... 26 F1 − F4 Conditions from Fuchs’s Criterion ...... 32 T Symbol representing a function y ...... 35 S Symbol representing a derivative y0 ...... 35 2 S Topological 2-sphere ...... 37 APP Algebraic PainlevéProperty ...... 37 d dz Ordinary (complex) differentiation ...... 41 ord f Order of a Puiseux series f ...... 43 Frac R Field of fractions of a domain R ...... 43 DX Sheaf of derivations on a curve X ...... 44 1 ΩX Sheaf of differential 1-forms on a curve X ...... 44 deg D Degree of a divisor D ...... 44 g(X) Genus of a curve X ...... 44 Ocp Completion of a local ring Op ...... 48 ⊗k Tensor product over a field k ...... 48 X 99K Y Rational map from X to Y ...... 49 X Algebraic surface ...... 49 ×k Fibred product of varieties over k ...... 51 ∇ Connection ...... 53 dgi Basis for the differential forms ...... 53 Hi ith cohomology functor ...... 54 Syms sth symmetric power ...... 54 L Line bundle ...... 54 Proj Proj-construction from algebraic geometry ...... 54

68 Figure B.1: A caricature of Paul Painlev´efrom 1932

69 70